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47 operations of 2nd degree expressions and formulas

The document discusses operations involving binomials and trinomials. It defines a binomial as a two-term polynomial of the form ax + b and a trinomial as a three-term polynomial of the form ax2 + bx + c. It states that the product of two binomials is a trinomial that can be found using the FOIL method: multiplying the first, outer, inner, and last terms. The FOIL method is demonstrated through examples multiplying binomial expressions. Expanding products involving negative binomials requires distributing the negative sign before using FOIL.

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Special Binomial Operations
A binomial is a two-term polynomial. Special Binomial Operations
A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b. Special Binomial Operations
A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Special Binomial Operations
A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c. Special Binomial Operations
A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c. The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + # Special Binomial Operations
A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c. The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + # Special Binomial Operations F: To get the x2-term, multiply the two Front x-terms of the binomials.
A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c. The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + # Special Binomial Operations F: To get the x2-term, multiply the two Front x-terms of the binomials. OI: To get the x-term, multiply the Outer and Inner pairs and combine the results.
A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c. The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + # Special Binomial Operations F: To get the x2-term, multiply the two Front x-terms of the binomials. OI: To get the x-term, multiply the Outer and Inner pairs and combine the results. L: To get the constant term, multiply the two Last constant terms.
A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c. The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + # Special Binomial Operations F: To get the x2-term, multiply the two Front x-terms of the binomials. OI: To get the x-term, multiply the Outer and Inner pairs and combine the results. L: To get the constant term, multiply the two Last constant terms. This is called the FOIL method.
A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c. The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + # Special Binomial Operations F: To get the x2-term, multiply the two Front x-terms of the binomials. OI: To get the x-term, multiply the Outer and Inner pairs and combine the results. L: To get the constant term, multiply the two Last constant terms. This is called the FOIL method. The FOIL method speeds up the multiplication of above binomial products and this will come in handy later.
Example A. Multiply using FOIL method. a. (x + 3)(x – 4) Special Binomial Operations
Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 Special Binomial Operations The front terms: x2-term
Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 Special Binomial Operations Outer pair: –4x
Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 Special Binomial Operations Inner pair: –4x + 3x
Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x Special Binomial Operations Outer Inner pairs: –4x + 3x = –x
Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 Special Binomial Operations The last terms: –12
Special Binomial Operations b. (3x + 4)(–2x + 5) Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12
Special Binomial Operations b. (3x + 4)(–2x + 5) = –6x2 The front terms: –6x2 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12
Special Binomial Operations b. (3x + 4)(–2x + 5) = –6x2 Outer pair: 15x Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12
Special Binomial Operations b. (3x + 4)(–2x + 5) = –6x2 Inner pair: 15x – 8x Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12
Special Binomial Operations b. (3x + 4)(–2x + 5) = –6x2 + 7x Outer and Inner pair: 15x – 8x = 7x Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12
Special Binomial Operations b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12
Special Binomial Operations b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 Expanding the negative of the binomial product requires extra care.
Special Binomial Operations b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product.
Special Binomial Operations b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product. Example B. Expand. a. – (3x – 4)(x + 5)
Special Binomial Operations b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product. Example B. Expand. a. – [(3x – 4)(x + 5)] Insert [ ]
Special Binomial Operations b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product. Example B. Expand. a. – [(3x – 4)(x + 5)] = – [ 3x2 + 15x – 4x – 20] Insert [ ] Expand
Special Binomial Operations b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product. Example B. Expand. a. – [(3x – 4)(x + 5)] = – [ 3x2 + 15x – 4x – 20] = – [ 3x2 + 11x – 20] Insert [ ] Expand
Special Binomial Operations b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product. Example B. Expand. a. – [(3x – 4)(x + 5)] = – [ 3x2 + 15x – 4x – 20] = – [ 3x2 + 11x – 20] = – 3x2 – 11x + 20 Insert [ ] Expand Remove [ ] and change signs.
Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL.
Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5)
Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) Distribute the sign.
Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 Distribute the sign. Expand
Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand
Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5)
Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5)
Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20)
Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20) = 2x2 + x – 15 – 3x2 – 11x + 20
Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20) = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5
Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20) = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 (2x – 5)(x +3) – (3x – 4)(x + 5)
Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20) = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 (2x – 5)(x +3) – [(3x – 4)(x + 5)] Insert brackets
Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20) = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 (2x – 5)(x +3) – [(3x – 4)(x + 5)] = 2x2 + x – 15 – [3x2 +11x – 20] Insert brackets Expand
Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20) = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 (2x – 5)(x +3) – [(3x – 4)(x + 5)] Insert brackets = 2x2 + x – 15 – [3x2 +11x – 20] = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 Expand Remove brackets and combine
Special Binomial Operations Another way to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20) = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 (2x – 5)(x +3) – [(3x – 4)(x + 5)] Insert brackets = 2x2 + x – 15 – [3x2 +11x – 20] = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 Expand Remove brackets and combine
Special Binomial Operations If the binomials are in x and y, then the products consist of the x2, xy and y2 terms.
Special Binomial Operations If the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, (#x + #y)(#x + #y) = #x2 + #xy + #y2
Special Binomial Operations If the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case.
Special Binomial Operations If the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, Example D. Expand. (3x – 4y)(x + 5y) (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case.
Special Binomial Operations If the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, Example D. Expand. (3x – 4y)(x + 5y) = 3x2 (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case. F
Special Binomial Operations If the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, Example D. Expand. (3x – 4y)(x + 5y) = 3x2 + 15xy – 4yx (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case. F OI
Special Binomial Operations If the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, Example D. Expand. (3x – 4y)(x + 5y) = 3x2 + 15xy – 4yx – 20y2 (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case. F OI L
Special Binomial Operations If the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, Example D. Expand. (3x – 4y)(x + 5y) = 3x2 + 15xy – 4yx – 20y2 = 3x2 + 11xy – 20y2 (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case.
Multiplication Formulas
There are some important patterns in multiplying expressions that it is worthwhile to memorize. Multiplication Formulas
The two binomials (A + B) and (A – B) are said to be the conjugate of each other. There are some important patterns in multiplying expressions that it is worthwhile to memorize. Multiplication Formulas
The two binomials (A + B) and (A – B) are said to be the conjugate of each other. For example, the conjugate of (3x + 2) is (3x – 2), There are some important patterns in multiplying expressions that it is worthwhile to memorize. Multiplication Formulas
The two binomials (A + B) and (A – B) are said to be the conjugate of each other. For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). There are some important patterns in multiplying expressions that it is worthwhile to memorize. Multiplication Formulas
The two binomials (A + B) and (A – B) are said to be the conjugate of each other. For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2). There are some important patterns in multiplying expressions that it is worthwhile to memorize. Multiplication Formulas
The two binomials (A + B) and (A – B) are said to be the conjugate of each other. There are some important patterns in multiplying expressions that it is worthwhile to memorize. I. Difference of Squares Formula Multiplication Formulas For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).
The two binomials (A + B) and (A – B) are said to be the conjugate of each other. There are some important patterns in multiplying expressions that it is worthwhile to memorize. I. Difference of Squares Formula (A + B)(A – B) Conjugate Product Multiplication Formulas For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).
The two binomials (A + B) and (A – B) are said to be the conjugate of each other. There are some important patterns in multiplying expressions that it is worthwhile to memorize. I. Difference of Squares Formula (A + B)(A – B) = A2 – B2 Conjugate Product Difference of Squares Multiplication Formulas For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).
The two binomials (A + B) and (A – B) are said to be the conjugate of each other. There are some important patterns in multiplying expressions that it is worthwhile to memorize. I. Difference of Squares Formula (A + B)(A – B) = A2 – B2 To verify this : (A + B)(A – B) Conjugate Product Difference of Squares Multiplication Formulas For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).
The two binomials (A + B) and (A – B) are said to be the conjugate of each other. There are some important patterns in multiplying expressions that it is worthwhile to memorize. I. Difference of Squares Formula (A + B)(A – B) = A2 – B2 To verify this : (A + B)(A – B) = A2 – AB + AB – B2 Conjugate Product Difference of Squares Multiplication Formulas For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).
The two binomials (A + B) and (A – B) are said to be the conjugate of each other. There are some important patterns in multiplying expressions that it is worthwhile to memorize. I. Difference of Squares Formula (A + B)(A – B) = A2 – B2 To verify this : (A + B)(A – B) = A2 – AB + AB – B2 = A2 – B2 Conjugate Product Difference of Squares Multiplication Formulas For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).
Multiplication Formulas Here are some examples of squaring:
Multiplication Formulas Here are some examples of squaring: (3x)2 =
Multiplication Formulas Here are some examples of squaring: (3x)2 = 9x2,
Multiplication Formulas Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 =
Multiplication Formulas Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2,
Multiplication Formulas Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2
Multiplication Formulas Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Multiplication Formulas Example E. Expand. a. (3x + 2)(3x – 2) Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Multiplication Formulas Example E. Expand. a. (3x + 2)(3x – 2) (A + B)(A – B) Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Multiplication Formulas Example E. Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 (A + B)(A – B) = A2 – B2 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Multiplication Formulas Example E. Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Multiplication Formulas Example E. Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Multiplication Formulas Example E. Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Multiplication Formulas Example E. Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Multiplication Formulas Example E. Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas
Multiplication Formulas Example E. Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2
Multiplication Formulas Example E. Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2
Multiplication Formulas Example E. Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2 We may check this easily by multiplying,
Multiplication Formulas Example E. Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2 We may check this easily by multiplying, (A + B)2 = (A + B)(A + B)
Multiplication Formulas Example E. Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2 We may check this easily by multiplying, (A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2
Multiplication Formulas Example E. Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2 We may check this easily by multiplying, (A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2 = A2 + 2AB + B2
Multiplication Formulas Example E. Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2 We may check this easily by multiplying, (A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2 = A2 + 2AB + B2 We say that “(A + B)2 is A2, B2, plus twice A*B”,
Multiplication Formulas Example E. Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2 We may check this easily by multiplying, (A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2 = A2 + 2AB + B2 We say that “(A + B)2 is A2, B2, plus twice A*B”, and “(A – B)2 is A2, B2, minus twice A*B”.
Example F. a. (3x + 4)2 Multiplication Formulas
Example F. a. (3x + 4)2 (A + B)2 Multiplication Formulas
Example F. a. (3x + 4)2 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas
Example F. a. (3x + 4)2 = (3x)2 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas
Example F. a. (3x + 4)2 = (3x)2 + 2(3x)(4) (A + B)2 = A2 + 2AB + B2 Multiplication Formulas
Example F. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas
Example F. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas
Example F. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2
Example F. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
Example F. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2
Example F. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas
Example F. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply.
Example F. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. Example G. Calculate. Use the conjugate formula. a. 51*49
Example F. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. Example G. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1)
Example F. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. Example G. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12
Example F. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. Example G. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499
Example F. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. Example G. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48
Example F. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. Example G. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22
Example F. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. Example G. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496
Example F. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. Example G. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496 c. 63*57 =
Example F. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. Example G. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496 c. 63*57 = (60 + 3)(60 – 3) = 602 – 32
Example F. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. Example G. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496 c. 63*57 = (60 + 3)(60 – 3) = 602 – 32 = 3,600 – 9 = 3,591
Example F. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. The conjugate formula (A + B)(A – B) = A2 – B2 may be used to multiply two numbers of the forms (A + B) and (A – B) where A2 and B2 can be calculated easily. Example G. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496 c. 63*57 = (60 + 3)(60 – 3) = 602 – 32 = 3,600 – 9 = 3,591
Multiplication Formulas We observe the algebraic patterns: (1 – r)(1 + r) = 1 – r2 (1 – r)(1 + r + r2) = 1 – r3 (1 – r)(1 + r + r2 + r3) = 1 – r4 (1 – r)(1 + r + r2 + r3 + r4) = 1 – r5 ... … (1 – r)(1 + r + r2 … + rn-1) = 1 – rn The Telescoping Products These are telescoping products, the products compress into two terms. In particular, we get the sum–of–powers formula: (1 – r)(1 + r + r2 … + rn-1) = 1 – rn 1 – r
Exercise. A. Calculate. Use the conjugate formula. Multiplication Formulas 1. 21*19 2. 31*29 3. 41*39 4. 71*69 5. 22*18 6. 32*28 7. 52*48 8. 73*67 B. Calculate. Use the squaring formula. 9. 212 10. 312 11. 392 12. 692 13. 982 14. 30½2 15. 100½2 16. 49½2 18. (x + 5)(x – 5) 19. (x – 7)(x + 7) 20. (2x + 3)(2x – 3) 21. (3x – 5)(3x + 5) C. Expand. 22. (7x + 2)(7x – 2) 23. (–7 + 3x )(–7 – 3x) 24. (–4x + 3)(–4x – 3) 25. (2x – 3y)(2x + 3y) 26. (4x – 5y)(5x + 5y) 27. (1 – 7y)(1 + 7y) 28. (5 – 3x)(5 + 3x) 29. (10 + 9x)(10 – 9x) 30. (x + 5)2 31. (x – 7)2 32. (2x + 3)2 33. (3x + 5y)2 34. (7x – 2y)2 35. (2x – h)2
B. Expand and simplify. Special Binomial Operations 1. (x + 5)(x + 7) 2. (x – 5)(x + 7) 3. (x + 5)(x – 7) 4. (x – 5)(x – 7) 5. (3x – 5)(2x + 4) 6. (–x + 5)(3x + 8) 7. (2x – 5)(2x + 5) 8. (3x + 7)(3x – 7) Exercise. A. Expand by FOIL method first. Then do them by inspection. 9. (–3x + 7)(4x + 3) 10. (–5x + 3)(3x – 4) 11. (2x – 5)(2x + 5) 12. (3x + 7)(3x – 7) 13. (9x + 4)(5x – 2) 14. (–5x + 3)(–3x + 1) 15. (5x – 1)(4x – 3) 16. (6x – 5)(–2x + 7) 17. (x + 5y)(x – 7y) 18. (x – 5y)(x – 7y) 19. (3x + 7y)(3x – 7y) 20. (–5x + 3y)(–3x + y) 21. –(2x – 5)(x + 3) 22. –(6x – 1)(3x – 4) 23. –(8x – 3)(2x + 1) 24. –(3x – 4)(4x – 3)
C. Expand and simplify. 25. (3x – 4)(x + 5) + (2x – 5)(x + 3) 26. (4x – 1)(2x – 5) + (x + 5)(x + 3) 27. (5x – 3)(x + 3) + (x + 5)(2x – 5) Special Binomial Operations 28. (3x – 4)(x + 5) – (2x – 5)(x + 3) 29. (4x – 4)(2x – 5) – (x + 5)(x + 3) 30. (5x – 3)(x + 3) – (x + 5)(2x – 5) 31. (2x – 7)(2x – 5) – (3x – 1)(2x + 3) 32. (3x – 1)(x – 7) – (x – 7)(3x + 1) 33. (2x – 3)(4x + 3) – (x + 2)(6x – 5) 34. (2x – 5)2 – (3x – 1)2 35. (x – 7)2 – (2x + 3)2 36. (4x + 3)2 – (6x – 5)2
Ex. D. Multiply the following monomials. 1. 3x2(–3x2) 11. 4x(3x – 5) – 9(6x – 7) Polynomial Operations 2. –3x2(8x5) 3. –5x2(–3x3) 4. –12( ) 6 –5x3 5. 24( x3) 8 –5 6. 6x2( ) 3 2x3 7. –15x4( x5) 5 –2 F. Expand and simplify. E. Fill in the degrees of the products. 8. #x(#x2 + # x + #) = #x? + #x? + #x? 9. #x2(#x4 + # x3 + #x2) = #x? + #x? + #x? 10. #x4(#x3 + # x2 + #x + #) = #x? + #x? + #x? + #x? 12. –x(2x + 7) + 3(4x – 2) 13. –3x(3x + 2) – 8x(7x – 5) 14. 5x(–5x + 9) + 6x(6x – 1) 15. 2x(–4x + 2) – 3x(2x – 1) – 3(4x – 2) 16. –4x(–7x + 9) – 2x(2x – 5) + 9(4x + 2)
18. (x + 5)(x + 7) Polynomial Operations G. Expand and simplify. (Use any method.) 19. (x – 5)(x + 7) 20. (x + 5)(x – 7) 21. (x – 5)(x – 7) 22. (3x – 5)(2x + 4) 23. (–x + 5)(3x + 8) 24. (2x – 5)(2x + 5) 25. (3x + 7)(3x – 7) 26. (3x2 – 5)(x – 6) 27. (8x – 2)(–4x2 – 7) 28. (2x – 7)(x2 – 3x + 9) 29. (5x + 3)(2x2 – x + 5) 38. (x – 1)(x + 1) 39. (x – 1)(x2 + x + 1) 40. (x – 1)(x3 + x2 + x + 1) 41. (x – 1)(x4 + x3 + x2 + x + 1) 42. What do you think the answer is for (x – 1)(x50 + x49 + …+ x2 + x + 1)? 30. (x – 1)(x – 1) 31. (x + 1)2 32. (2x – 3)2 33. (5x + 4)2 34. 2x(2x – 1)(3x + 2) 35. 4x(3x – 2)(2x + 3) 36. (x – 5)(2x – 1)(3x + 2) 37. (2x + 1)(3x + 1)(x – 2)

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47 operations of 2nd degree expressions and formulas

  • 1.
  • 2.
    A binomial isa two-term polynomial. Special Binomial Operations
  • 3.
    A binomial isa two-term polynomial. Usually we use the term for expressions of the form ax + b. Special Binomial Operations
  • 4.
    A binomial isa two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Special Binomial Operations
  • 5.
    A binomial isa two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c. Special Binomial Operations
  • 6.
    A binomial isa two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c. The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + # Special Binomial Operations
  • 7.
    A binomial isa two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c. The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + # Special Binomial Operations F: To get the x2-term, multiply the two Front x-terms of the binomials.
  • 8.
    A binomial isa two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c. The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + # Special Binomial Operations F: To get the x2-term, multiply the two Front x-terms of the binomials. OI: To get the x-term, multiply the Outer and Inner pairs and combine the results.
  • 9.
    A binomial isa two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c. The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + # Special Binomial Operations F: To get the x2-term, multiply the two Front x-terms of the binomials. OI: To get the x-term, multiply the Outer and Inner pairs and combine the results. L: To get the constant term, multiply the two Last constant terms.
  • 10.
    A binomial isa two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c. The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + # Special Binomial Operations F: To get the x2-term, multiply the two Front x-terms of the binomials. OI: To get the x-term, multiply the Outer and Inner pairs and combine the results. L: To get the constant term, multiply the two Last constant terms. This is called the FOIL method.
  • 11.
    A binomial isa two-term polynomial. Usually we use the term for expressions of the form ax + b. A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c. The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + # Special Binomial Operations F: To get the x2-term, multiply the two Front x-terms of the binomials. OI: To get the x-term, multiply the Outer and Inner pairs and combine the results. L: To get the constant term, multiply the two Last constant terms. This is called the FOIL method. The FOIL method speeds up the multiplication of above binomial products and this will come in handy later.
  • 12.
    Example A. Multiplyusing FOIL method. a. (x + 3)(x – 4) Special Binomial Operations
  • 13.
    Example A. Multiplyusing FOIL method. a. (x + 3)(x – 4) = x2 Special Binomial Operations The front terms: x2-term
  • 14.
    Example A. Multiplyusing FOIL method. a. (x + 3)(x – 4) = x2 Special Binomial Operations Outer pair: –4x
  • 15.
    Example A. Multiplyusing FOIL method. a. (x + 3)(x – 4) = x2 Special Binomial Operations Inner pair: –4x + 3x
  • 16.
    Example A. Multiplyusing FOIL method. a. (x + 3)(x – 4) = x2 – x Special Binomial Operations Outer Inner pairs: –4x + 3x = –x
  • 17.
    Example A. Multiplyusing FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 Special Binomial Operations The last terms: –12
  • 18.
    Special Binomial Operations b.(3x + 4)(–2x + 5) Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12
  • 19.
    Special Binomial Operations b.(3x + 4)(–2x + 5) = –6x2 The front terms: –6x2 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12
  • 20.
    Special Binomial Operations b.(3x + 4)(–2x + 5) = –6x2 Outer pair: 15x Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12
  • 21.
    Special Binomial Operations b.(3x + 4)(–2x + 5) = –6x2 Inner pair: 15x – 8x Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12
  • 22.
    Special Binomial Operations b.(3x + 4)(–2x + 5) = –6x2 + 7x Outer and Inner pair: 15x – 8x = 7x Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: –12
  • 23.
    Special Binomial Operations b.(3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12
  • 24.
    Special Binomial Operations b.(3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 Expanding the negative of the binomial product requires extra care.
  • 25.
    Special Binomial Operations b.(3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product.
  • 26.
    Special Binomial Operations b.(3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product. Example B. Expand. a. – (3x – 4)(x + 5)
  • 27.
    Special Binomial Operations b.(3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product. Example B. Expand. a. – [(3x – 4)(x + 5)] Insert [ ]
  • 28.
    Special Binomial Operations b.(3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product. Example B. Expand. a. – [(3x – 4)(x + 5)] = – [ 3x2 + 15x – 4x – 20] Insert [ ] Expand
  • 29.
    Special Binomial Operations b.(3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product. Example B. Expand. a. – [(3x – 4)(x + 5)] = – [ 3x2 + 15x – 4x – 20] = – [ 3x2 + 11x – 20] Insert [ ] Expand
  • 30.
    Special Binomial Operations b.(3x + 4)(–2x + 5) = –6x2 + 7x + 20 Example A. Multiply using FOIL method. a. (x + 3)(x – 4) = x2 – x – 12 The last terms: 20 The last terms: –12 Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product. Example B. Expand. a. – [(3x – 4)(x + 5)] = – [ 3x2 + 15x – 4x – 20] = – [ 3x2 + 11x – 20] = – 3x2 – 11x + 20 Insert [ ] Expand Remove [ ] and change signs.
  • 31.
    Special Binomial Operations Anotherway to do this is to distribute the negative sign into the first binomial then FOIL.
  • 32.
    Special Binomial Operations Anotherway to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5)
  • 33.
    Special Binomial Operations Anotherway to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) Distribute the sign.
  • 34.
    Special Binomial Operations Anotherway to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 Distribute the sign. Expand
  • 35.
    Special Binomial Operations Anotherway to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand
  • 36.
    Special Binomial Operations Anotherway to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5)
  • 37.
    Special Binomial Operations Anotherway to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5)
  • 38.
    Special Binomial Operations Anotherway to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20)
  • 39.
    Special Binomial Operations Anotherway to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20) = 2x2 + x – 15 – 3x2 – 11x + 20
  • 40.
    Special Binomial Operations Anotherway to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20) = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5
  • 41.
    Special Binomial Operations Anotherway to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20) = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 (2x – 5)(x +3) – (3x – 4)(x + 5)
  • 42.
    Special Binomial Operations Anotherway to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20) = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 (2x – 5)(x +3) – [(3x – 4)(x + 5)] Insert brackets
  • 43.
    Special Binomial Operations Anotherway to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20) = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 (2x – 5)(x +3) – [(3x – 4)(x + 5)] = 2x2 + x – 15 – [3x2 +11x – 20] Insert brackets Expand
  • 44.
    Special Binomial Operations Anotherway to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20) = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 (2x – 5)(x +3) – [(3x – 4)(x + 5)] Insert brackets = 2x2 + x – 15 – [3x2 +11x – 20] = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 Expand Remove brackets and combine
  • 45.
    Special Binomial Operations Anotherway to do this is to distribute the negative sign into the first binomial then FOIL. Example C. Expand. a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20 Distribute the sign. Expand b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5) = (2x – 5)(x +3) + (–3x + 4)(x + 5) = 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20) = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 (2x – 5)(x +3) – [(3x – 4)(x + 5)] Insert brackets = 2x2 + x – 15 – [3x2 +11x – 20] = 2x2 + x – 15 – 3x2 – 11x + 20 = –x2 – 10x + 5 Expand Remove brackets and combine
  • 46.
    Special Binomial Operations Ifthe binomials are in x and y, then the products consist of the x2, xy and y2 terms.
  • 47.
    Special Binomial Operations Ifthe binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, (#x + #y)(#x + #y) = #x2 + #xy + #y2
  • 48.
    Special Binomial Operations Ifthe binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case.
  • 49.
    Special Binomial Operations Ifthe binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, Example D. Expand. (3x – 4y)(x + 5y) (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case.
  • 50.
    Special Binomial Operations Ifthe binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, Example D. Expand. (3x – 4y)(x + 5y) = 3x2 (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case. F
  • 51.
    Special Binomial Operations Ifthe binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, Example D. Expand. (3x – 4y)(x + 5y) = 3x2 + 15xy – 4yx (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case. F OI
  • 52.
    Special Binomial Operations Ifthe binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, Example D. Expand. (3x – 4y)(x + 5y) = 3x2 + 15xy – 4yx – 20y2 (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case. F OI L
  • 53.
    Special Binomial Operations Ifthe binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is, Example D. Expand. (3x – 4y)(x + 5y) = 3x2 + 15xy – 4yx – 20y2 = 3x2 + 11xy – 20y2 (#x + #y)(#x + #y) = #x2 + #xy + #y2 The FOIL method is still applicable in this case.
  • 54.
  • 55.
    There are someimportant patterns in multiplying expressions that it is worthwhile to memorize. Multiplication Formulas
  • 56.
    The two binomials(A + B) and (A – B) are said to be the conjugate of each other. There are some important patterns in multiplying expressions that it is worthwhile to memorize. Multiplication Formulas
  • 57.
    The two binomials(A + B) and (A – B) are said to be the conjugate of each other. For example, the conjugate of (3x + 2) is (3x – 2), There are some important patterns in multiplying expressions that it is worthwhile to memorize. Multiplication Formulas
  • 58.
    The two binomials(A + B) and (A – B) are said to be the conjugate of each other. For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). There are some important patterns in multiplying expressions that it is worthwhile to memorize. Multiplication Formulas
  • 59.
    The two binomials(A + B) and (A – B) are said to be the conjugate of each other. For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2). There are some important patterns in multiplying expressions that it is worthwhile to memorize. Multiplication Formulas
  • 60.
    The two binomials(A + B) and (A – B) are said to be the conjugate of each other. There are some important patterns in multiplying expressions that it is worthwhile to memorize. I. Difference of Squares Formula Multiplication Formulas For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).
  • 61.
    The two binomials(A + B) and (A – B) are said to be the conjugate of each other. There are some important patterns in multiplying expressions that it is worthwhile to memorize. I. Difference of Squares Formula (A + B)(A – B) Conjugate Product Multiplication Formulas For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).
  • 62.
    The two binomials(A + B) and (A – B) are said to be the conjugate of each other. There are some important patterns in multiplying expressions that it is worthwhile to memorize. I. Difference of Squares Formula (A + B)(A – B) = A2 – B2 Conjugate Product Difference of Squares Multiplication Formulas For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).
  • 63.
    The two binomials(A + B) and (A – B) are said to be the conjugate of each other. There are some important patterns in multiplying expressions that it is worthwhile to memorize. I. Difference of Squares Formula (A + B)(A – B) = A2 – B2 To verify this : (A + B)(A – B) Conjugate Product Difference of Squares Multiplication Formulas For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).
  • 64.
    The two binomials(A + B) and (A – B) are said to be the conjugate of each other. There are some important patterns in multiplying expressions that it is worthwhile to memorize. I. Difference of Squares Formula (A + B)(A – B) = A2 – B2 To verify this : (A + B)(A – B) = A2 – AB + AB – B2 Conjugate Product Difference of Squares Multiplication Formulas For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).
  • 65.
    The two binomials(A + B) and (A – B) are said to be the conjugate of each other. There are some important patterns in multiplying expressions that it is worthwhile to memorize. I. Difference of Squares Formula (A + B)(A – B) = A2 – B2 To verify this : (A + B)(A – B) = A2 – AB + AB – B2 = A2 – B2 Conjugate Product Difference of Squares Multiplication Formulas For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).
  • 66.
    Multiplication Formulas Here aresome examples of squaring:
  • 67.
    Multiplication Formulas Here aresome examples of squaring: (3x)2 =
  • 68.
    Multiplication Formulas Here aresome examples of squaring: (3x)2 = 9x2,
  • 69.
    Multiplication Formulas Here aresome examples of squaring: (3x)2 = 9x2, (2xy)2 =
  • 70.
    Multiplication Formulas Here aresome examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2,
  • 71.
    Multiplication Formulas Here aresome examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2
  • 72.
    Multiplication Formulas Here aresome examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
  • 73.
    Multiplication Formulas Example E.Expand. a. (3x + 2)(3x – 2) Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
  • 74.
    Multiplication Formulas Example E.Expand. a. (3x + 2)(3x – 2) (A + B)(A – B) Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
  • 75.
    Multiplication Formulas Example E.Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 (A + B)(A – B) = A2 – B2 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
  • 76.
    Multiplication Formulas Example E.Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
  • 77.
    Multiplication Formulas Example E.Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
  • 78.
    Multiplication Formulas Example E.Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
  • 79.
    Multiplication Formulas Example E.Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
  • 80.
    Multiplication Formulas Example E.Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas
  • 81.
    Multiplication Formulas Example E.Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2
  • 82.
    Multiplication Formulas Example E.Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2
  • 83.
    Multiplication Formulas Example E.Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2 We may check this easily by multiplying,
  • 84.
    Multiplication Formulas Example E.Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2 We may check this easily by multiplying, (A + B)2 = (A + B)(A + B)
  • 85.
    Multiplication Formulas Example E.Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2 We may check this easily by multiplying, (A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2
  • 86.
    Multiplication Formulas Example E.Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2 We may check this easily by multiplying, (A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2 = A2 + 2AB + B2
  • 87.
    Multiplication Formulas Example E.Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2 We may check this easily by multiplying, (A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2 = A2 + 2AB + B2 We say that “(A + B)2 is A2, B2, plus twice A*B”,
  • 88.
    Multiplication Formulas Example E.Expand. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2 We may check this easily by multiplying, (A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2 = A2 + 2AB + B2 We say that “(A + B)2 is A2, B2, plus twice A*B”, and “(A – B)2 is A2, B2, minus twice A*B”.
  • 89.
    Example F. a. (3x+ 4)2 Multiplication Formulas
  • 90.
    Example F. a. (3x+ 4)2 (A + B)2 Multiplication Formulas
  • 91.
    Example F. a. (3x+ 4)2 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas
  • 92.
    Example F. a. (3x+ 4)2 = (3x)2 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas
  • 93.
    Example F. a. (3x+ 4)2 = (3x)2 + 2(3x)(4) (A + B)2 = A2 + 2AB + B2 Multiplication Formulas
  • 94.
    Example F. a. (3x+ 4)2 = (3x)2 + 2(3x)(4) + 42 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas
  • 95.
    Example F. a. (3x+ 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas
  • 96.
    Example F. a. (3x+ 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2
  • 97.
    Example F. a. (3x+ 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
  • 98.
    Example F. a. (3x+ 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2
  • 99.
    Example F. a. (3x+ 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas
  • 100.
    Example F. a. (3x+ 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply.
  • 101.
    Example F. a. (3x+ 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. Example G. Calculate. Use the conjugate formula. a. 51*49
  • 102.
    Example F. a. (3x+ 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. Example G. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1)
  • 103.
    Example F. a. (3x+ 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. Example G. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12
  • 104.
    Example F. a. (3x+ 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. Example G. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499
  • 105.
    Example F. a. (3x+ 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. Example G. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48
  • 106.
    Example F. a. (3x+ 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. Example G. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22
  • 107.
    Example F. a. (3x+ 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. Example G. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496
  • 108.
    Example F. a. (3x+ 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. Example G. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496 c. 63*57 =
  • 109.
    Example F. a. (3x+ 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. Example G. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496 c. 63*57 = (60 + 3)(60 – 3) = 602 – 32
  • 110.
    Example F. a. (3x+ 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. Example G. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496 c. 63*57 = (60 + 3)(60 – 3) = 602 – 32 = 3,600 – 9 = 3,591
  • 111.
    Example F. a. (3x+ 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply. The conjugate formula (A + B)(A – B) = A2 – B2 may be used to multiply two numbers of the forms (A + B) and (A – B) where A2 and B2 can be calculated easily. Example G. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496 c. 63*57 = (60 + 3)(60 – 3) = 602 – 32 = 3,600 – 9 = 3,591
  • 112.
    Multiplication Formulas We observethe algebraic patterns: (1 – r)(1 + r) = 1 – r2 (1 – r)(1 + r + r2) = 1 – r3 (1 – r)(1 + r + r2 + r3) = 1 – r4 (1 – r)(1 + r + r2 + r3 + r4) = 1 – r5 ... … (1 – r)(1 + r + r2 … + rn-1) = 1 – rn The Telescoping Products These are telescoping products, the products compress into two terms. In particular, we get the sum–of–powers formula: (1 – r)(1 + r + r2 … + rn-1) = 1 – rn 1 – r
  • 113.
    Exercise. A. Calculate.Use the conjugate formula. Multiplication Formulas 1. 21*19 2. 31*29 3. 41*39 4. 71*69 5. 22*18 6. 32*28 7. 52*48 8. 73*67 B. Calculate. Use the squaring formula. 9. 212 10. 312 11. 392 12. 692 13. 982 14. 30½2 15. 100½2 16. 49½2 18. (x + 5)(x – 5) 19. (x – 7)(x + 7) 20. (2x + 3)(2x – 3) 21. (3x – 5)(3x + 5) C. Expand. 22. (7x + 2)(7x – 2) 23. (–7 + 3x )(–7 – 3x) 24. (–4x + 3)(–4x – 3) 25. (2x – 3y)(2x + 3y) 26. (4x – 5y)(5x + 5y) 27. (1 – 7y)(1 + 7y) 28. (5 – 3x)(5 + 3x) 29. (10 + 9x)(10 – 9x) 30. (x + 5)2 31. (x – 7)2 32. (2x + 3)2 33. (3x + 5y)2 34. (7x – 2y)2 35. (2x – h)2
  • 114.
    B. Expand andsimplify. Special Binomial Operations 1. (x + 5)(x + 7) 2. (x – 5)(x + 7) 3. (x + 5)(x – 7) 4. (x – 5)(x – 7) 5. (3x – 5)(2x + 4) 6. (–x + 5)(3x + 8) 7. (2x – 5)(2x + 5) 8. (3x + 7)(3x – 7) Exercise. A. Expand by FOIL method first. Then do them by inspection. 9. (–3x + 7)(4x + 3) 10. (–5x + 3)(3x – 4) 11. (2x – 5)(2x + 5) 12. (3x + 7)(3x – 7) 13. (9x + 4)(5x – 2) 14. (–5x + 3)(–3x + 1) 15. (5x – 1)(4x – 3) 16. (6x – 5)(–2x + 7) 17. (x + 5y)(x – 7y) 18. (x – 5y)(x – 7y) 19. (3x + 7y)(3x – 7y) 20. (–5x + 3y)(–3x + y) 21. –(2x – 5)(x + 3) 22. –(6x – 1)(3x – 4) 23. –(8x – 3)(2x + 1) 24. –(3x – 4)(4x – 3)
  • 115.
    C. Expand andsimplify. 25. (3x – 4)(x + 5) + (2x – 5)(x + 3) 26. (4x – 1)(2x – 5) + (x + 5)(x + 3) 27. (5x – 3)(x + 3) + (x + 5)(2x – 5) Special Binomial Operations 28. (3x – 4)(x + 5) – (2x – 5)(x + 3) 29. (4x – 4)(2x – 5) – (x + 5)(x + 3) 30. (5x – 3)(x + 3) – (x + 5)(2x – 5) 31. (2x – 7)(2x – 5) – (3x – 1)(2x + 3) 32. (3x – 1)(x – 7) – (x – 7)(3x + 1) 33. (2x – 3)(4x + 3) – (x + 2)(6x – 5) 34. (2x – 5)2 – (3x – 1)2 35. (x – 7)2 – (2x + 3)2 36. (4x + 3)2 – (6x – 5)2
  • 116.
    Ex. D. Multiplythe following monomials. 1. 3x2(–3x2) 11. 4x(3x – 5) – 9(6x – 7) Polynomial Operations 2. –3x2(8x5) 3. –5x2(–3x3) 4. –12( ) 6 –5x3 5. 24( x3) 8 –5 6. 6x2( ) 3 2x3 7. –15x4( x5) 5 –2 F. Expand and simplify. E. Fill in the degrees of the products. 8. #x(#x2 + # x + #) = #x? + #x? + #x? 9. #x2(#x4 + # x3 + #x2) = #x? + #x? + #x? 10. #x4(#x3 + # x2 + #x + #) = #x? + #x? + #x? + #x? 12. –x(2x + 7) + 3(4x – 2) 13. –3x(3x + 2) – 8x(7x – 5) 14. 5x(–5x + 9) + 6x(6x – 1) 15. 2x(–4x + 2) – 3x(2x – 1) – 3(4x – 2) 16. –4x(–7x + 9) – 2x(2x – 5) + 9(4x + 2)
  • 117.
    18. (x +5)(x + 7) Polynomial Operations G. Expand and simplify. (Use any method.) 19. (x – 5)(x + 7) 20. (x + 5)(x – 7) 21. (x – 5)(x – 7) 22. (3x – 5)(2x + 4) 23. (–x + 5)(3x + 8) 24. (2x – 5)(2x + 5) 25. (3x + 7)(3x – 7) 26. (3x2 – 5)(x – 6) 27. (8x – 2)(–4x2 – 7) 28. (2x – 7)(x2 – 3x + 9) 29. (5x + 3)(2x2 – x + 5) 38. (x – 1)(x + 1) 39. (x – 1)(x2 + x + 1) 40. (x – 1)(x3 + x2 + x + 1) 41. (x – 1)(x4 + x3 + x2 + x + 1) 42. What do you think the answer is for (x – 1)(x50 + x49 + …+ x2 + x + 1)? 30. (x – 1)(x – 1) 31. (x + 1)2 32. (2x – 3)2 33. (5x + 4)2 34. 2x(2x – 1)(3x + 2) 35. 4x(3x – 2)(2x + 3) 36. (x – 5)(2x – 1)(3x + 2) 37. (2x + 1)(3x + 1)(x – 2)