March 6, 2014
- 2. Warm-Up: The speed of light is 186,000 miles per second. How far can light travel in one minute? Write your answer in scientific notation.
- 3. Review From Yesterday: Like terms have the same exponent to the same degree. When adding or subtracting, only like terms can be combined. When polynomials have more than one variable, the same rules apply. For example: (xz + 5x²z – x) + (x+ 5z) (x + z) + (zx+ z²x) (a + 5ba) + (3ba+ a²) (3xyz - xyz + zx) + (3zyx+ 1) Like terms have the same exponent to the same degree....... But order does not matter!
- 4. Multiplying Polynomials: Let's begin by multiplying a monomial by a monomial. = 2x7 If the bases (x) are the same, we add the exponents 2x7y (2x ) • (yx ) = Now multiply a monomial by a binomial 3 4 Once more: x(7x2 + 4y) = 7x3 + 4xy 7x3 + 4x When multiplying polynomials, each term is multiplied by every other term. Now we look at multiplying a binomial by a binomial.
- 5. Multiplying Binomials Binomials Method #1: The Box Method (x + 4)(x + 2) *Reminder: When multiplying, add the exponents if bases are alike
- 6. Multiplying Binomials Use the Box Method: = x2 -3x +4x -12 =
- 7. Multiplying Binomials = = However, the More Common Method for solving binomials is...
- 9. F.O.I.L. If we perform our distribution in this order, (x + 1)(x + 2) = x (x + 2) + 1 (x + 2) a useful pattern emerges. Distributing produces the sum of these four multiplications. First + Outer + Inner + Last "F.O.I.L" for short. (x + 1) (x + 2) = x ( x + 2 ) + 1 ( x + 2 ) x2 + 2x + x + 2 x2 + 3x + 2
- 10. Multiplying Binomials Mentally Find the pattern (x + 2)(x + 1) x2 + x + 2x + 2 x2 + 3x + 2 (x + 3)(x + 2) x2 + 2x + 3x + 6 x2 + 5x + 6 (x + 4)(x + 3) x2 + 3x + 4x + 12 x2 + 7x + 12 (x + 5)(x + 4) x2 + 9x + 20 x2 + 4x + 5x + 20 (x + 6)(x + 5) x2 + 5x + 6x + 30 x2 + 11x + 30 There are lots of patterns here, but this one enables us to multiply (x + a)(x + b) = x2 + x(a + b) + ab binomials mentally. The middle term of the answer is the sum of the binomial's last terms and the last term in the answer is the product of the binomial's last terms.
- 11. Positive and Negative All of the binomials we have multiplied so far have been sums of positive numbers. What happens if one of the terms is negative? Example 1: (x + 4)(x - 3) 1. The last term will be negative, because a positive times a negative is negative. 2. The middle term in this example will be positive, because 4 + (- 3) = 1. (x + 4)(x - 3) = x2 + x - 12 Example 2: (x - 4)(x + 3) 1. The last term will still be negative, because a positive times a negative is negative. 2. But the middle term in this example will be negative, because (- 4) + 3 = - 1. (x - 4)(x + 3) = x2 - x - 12
- 12. Two Negatives What happens if the second term in both binomials is negative? Example: (x - 4)(x - 3) 1. The last term will be positive, because a negative times a negative is positive. 2. The middle term will be negative, because a negative plus a negative is negative. (x - 4)(x - 3) = x2 -7x +12 Compare this result to what happens when both terms are positive: (x + 4)(x + 3) = x2 +7x +12 Both signs the same: last term positive middle term the same
- 13. Sign Summary Middle Term Last Term (x + 4)(x + 3) positive positive (x - 4)(x + 3) negative negative (x + 4)(x - 3) positive negative (x - 4)(x - 3) negative positive Which term is larger doesn't matter when both signs are the same, but it does when the signs are different.
- 14. Remember, F.O.I.L can be used when multiplying a binomial by another binomial.