1.5 Factoring Polynomials
- 1. 1.5 Factoring Polynomials Chapter 1 Prerequisites
- 2. Concepts and Objectives ⚫ Objectives for this section: ⚫ Factor the greatest common factor of a polynomial. ⚫ Factor a trinomial. ⚫ Factor by grouping. ⚫ Factor a perfect square trinomial. ⚫ Factor a difference of squares. ⚫ Factor the sum and difference of cubes. ⚫ Factor expressions using fractional or negative exponents.
- 3. Factoring Polynomials ⚫ The process of finding polynomials whose product equals a given polynomial is called factoring. ⚫ For example, since 4x + 12 = 4(x + 3), both 4 and x + 3 are called factors of 4x + 12. ⚫ A polynomial that cannot be written as a product of two polynomials of lower degree is a prime polynomial. ⚫ One nice aspect of this process is that it has a built-in check: whatever factors you come up with, you should be able to multiply them and get your starting expression.
- 4. Factoring Out the GCF Factor out the greatest common factor from each polynomial: ⚫ ⚫ ⚫ 5 2 9y y + 2 6 8 12 x t xt t + + ( ) ( ) ( ) 3 2 14 1 28 1 7 1 m m m + − + − +
- 5. Factoring Out the GCF Factor out the greatest common factor from each polynomial: ⚫ GCF: y2 ⚫ GCF: 2t ⚫ GCF: 7m + 1 5 2 9y y + 2 6 8 12 x t xt t + + ( ) ( ) ( ) 3 2 14 1 28 1 7 1 m m m + − + − + ( ) 3 2 9 1 y y + ( ) 2 6 2 3 4 x t x + + ( ) ( ) ( ) 2 4 7 1 2 1 1 1 m m m + − + − +
- 6. Factoring Out the GCF (cont.) We can clean up that last problem just a little more: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) + + − + − + + + − + − + + + − − − + − 2 2 2 2 7 1 2 1 4 1 1 7 1 2 2 1 4 1 1 7 1 2 4 2 4 4 1 7 1 2 3 m m m m m m m m m m m m m
- 7. Factoring by Grouping ⚫ When a polynomial has more than three terms, it can sometimes be factored using factoring by grouping. ⚫ For example, to factor ax + ay + 6x + 6y, group the terms so that each group has a common factor. ( ) ( ) ( ) ( ) ( )( ) 6 6 6 6 6 6 ax ay x y ax ay x y a x y x y x y a + + + = + + + = + + + = + + x + y is the GCF of the expression above.
- 8. Factoring by Grouping (cont.) Examples: Factor each polynomial by grouping. ⚫ ⚫ 2 2 7 3 21 mp m p + + + 2 2 2 2 y az z ay + − −
- 9. Factoring by Grouping (cont.) Examples: Factor each polynomial by grouping. ⚫ ⚫ ( ) ( ) ( )( ) 2 2 2 2 2 7 3 21 7 3 7 7 3 mp m p m p p p m + + + = + + + = + + 2 2 2 2 y az z ay + − −
- 10. Factoring by Grouping (cont.) Examples: Factor each polynomial by grouping. ⚫ ⚫ ( ) ( ) ( )( ) 2 2 2 2 2 7 3 21 7 3 7 7 3 mp m p m p p p m + + + = + + + = + + ( ) ( ) ( ) ( ) ( )( ) + − − = − − + = − + − + = − − − = − − 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 y az z ay y z ay az y z ay az y z a y z y z a You can rearrange the terms to make grouping easier.
- 11. Factoring by Grouping (cont.) Examples: Factor each polynomial by grouping. ⚫ 3 2 4 2 2 1 x x x + − −
- 12. Factoring by Grouping (cont.) Examples: Factor each polynomial by grouping. ⚫ ( ) ( ) ( ) ( ) ( )( ) 3 2 3 2 2 2 4 2 2 1 4 2 2 1 2 2 1 1 2 1 2 1 2 1 x x x x x x x x x x x − + + − − = + = − = + + + −
- 13. Factoring Trinomials If you have an expression of the form ax2 +bx + c, you can use one of the following methods to factor it: ⚫ X-method (a = 1): If a = 1, this is the simplest method to use. Find two numbers that multiply to c and add up to b. These two numbers will create your factors. ⚫ Example: Factor x2 ‒ 5x ‒ 14. ‒14 ‒7 2 ‒5 ( )( ) 2 5 14 7 2 x x x x − − = − + c b
- 14. Factoring Trinomials (cont.) ⚫ Reverse box: If a is greater than 1, you can use the previous X method to split the middle term (ac goes on top) and use either grouping or the box method, and then find the GCF of each column and row. ⚫ Example: Factor Now, find the GCF of each line. − − 2 4 5 6 y y ‒24 ‒8 3 5 4y2 ‒8y 3y ‒6 ac b
- 15. Factoring Trinomials (cont.) ⚫ Reverse box: If a is greater than 1, you can use the previous X method to split the middle term (ac goes on top) and use either grouping or the box method, and then find the GCF of each column and row. ⚫ Example: Factor − − 2 4 5 6 y y ‒24 ‒8 3 5 y ‒2 4y 4y2 ‒8y 3 3y ‒6 ( )( ) − − = + − 2 4 5 6 4 3 2 y y y y
- 16. Factoring Trinomials (cont.) ⚫ Grouping: This method is about the same as the Reverse Box, except that it is not in a graphic format. ⚫ Example: Factor 2 2 6 x x − − ‒12 ‒4 3 ‒1 ( ) ( ) ( ) ( ) ( )( ) 2 2 2 2 6 2 6 2 4 3 6 2 2 3 2 2 2 3 4 3 x x x x x x x x x x x x x − − = − = − + − = = + − − − + − +
- 17. Factoring Trinomials (cont.) ⚫ Mustang: This method is named after the mnemonic “My Father Drives A Red Mustang”, where the letters stand for: M Multiply a and c. F Find factors using the X method. Set up ( ). DA Divide the factors by a if necessary. R Reduce any fractions. M Move any denominators to the front of the variable.
- 18. Factoring Trinomials (cont.) ⚫ Example: Factor 2 5 7 6 x x + − M Multiply (5)(‒6) = ‒30 F Find factors: DA Divide by a R Reduce fractions M Move the denominator ‒30 10 ‒3 7 ( )( ) 10 3 x x + − 10 3 5 5 x x + − ( ) 3 2 5 x x + − ( )( ) 2 5 3 x x + −
- 19. Perfect Square Trinomials ⚫ We can use the reverse of the special patterns we saw last class to quickly factor perfect square trinomials if we can recognize the pattern. ⚫ If you encounter a trinomial which fits this pattern, you can quickly factor it by taking the square roots of the first and last term. ⚫ Make sure to verify that the middle term is ! ( ) 2 2 2 2 a ab b a b + = 2ab
- 20. Perfect Square Trinomials ⚫ Example: Factor 9x2 ‒ 12x + 4
- 21. Perfect Square Trinomials ⚫ Example: Factor 9x2 ‒ 12x + 4 The first thing to notice is that the first and last terms are perfect squares, and that the middle term is two times the product of the square roots. To factor this, put the two square roots together, along with whatever the sign is between the first and second term. 2 9 3 and 4 2 x x = = ( )( ) 2 3 4 12 x x = ( ) 2 2 9 12 4 3 2 x x x + = − −
- 22. Perfect Square Trinomials Examples: Factor the following ⚫ ⚫ ⚫ 2 2 16 40 25 p pq q − + 2 2 36 84 49 x y xy + + 2 81 90 25 a a − +
- 23. Perfect Square Trinomials Examples: Factor the following ⚫ ⚫ ⚫ ( ) ( )( ) ( ) ( ) 2 2 2 2 2 16 40 25 4 2 4 5 5 4 5 p pq q p p q q p q − + = − + = − ( ) ( )( ) ( ) 2 2 2 2 2 36 84 49 6 2 6 7 7 6 7 x y xy xy xy xy + + = + + = + ( ) ( )( ) ( ) 2 2 2 2 81 90 25 9 2 9 5 5 9 5 a a a a a − + = − + = −
- 24. Factoring Binomials ⚫ If you are asked to factor a binomial (2 terms), check first for common factors, then check to see if it fits one of the following patterns: ⚫ Note: There is no factoring pattern for a sum of squares (a2 + b2) in the real number system. Difference of Squares a2 ‒ b2 = a + ba ‒ b Sum/Diff. of Cubes ( )( ) 3 3 2 2 a b a b a ab b = +
- 25. Factoring Binomials (cont.) Examples ⚫ Factor ⚫ Factor ⚫ Factor 2 4 81 x − 3 27 x − 3 3 24 x + ( ) ( )( ) 2 2 2 9 2 9 2 9 x x x = − = − + ( )( ) 3 3 2 3 3 3 9 x x x x = + − = − + ( ) ( ) ( )( ) 3 3 3 2 3 8 3 2 3 2 2 4 x x x x x = + = + = + − +
- 26. Factoring Binomials (cont.) EVERY TIME YOU DO THIS: A KITTEN DIES ( ) 2 2 2 x y x y + = + Remember:
- 27. Factoring Rational Exponents ⚫ When factoring expressions with negative or rational exponents, factor out the smallest power of the variable or variable expression. ⚫ Example: ⚫ ⚫ 2 3 12 8 x x − − − 1/2 3/2 4 3 m m +
- 28. Factoring Rational Exponents ⚫ When factoring expressions with negative or rational exponents, just as before, factor out the least power of the variable or variable expression. ⚫ Example: ⚫ ⚫ 2 3 12 8 x x − − − 1/2 3/2 4 3 m m + ( ) ( ) ( ) 2 3 3 3 3 4 3 2 x x x − − − − − − − = − ( ) 3 4 3 2 x x − = −
- 29. Factoring Rational Exponents ⚫ When factoring expressions with negative or rational exponents, factor out the smallest power of the variable or variable expression. ⚫ Examples: ⚫ ⚫ 2 3 12 8 x x − − − 1/2 3/2 4 3 m m + ( ) ( ) ( ) 2 3 3 3 3 4 3 2 x x x − − − − − − − = − ( ) 3 4 3 2 x x − = − ( ) 1/2 1/2 1/2 3/2 1/2 4 3 m m m − − = + ( ) 1/2 4 3 m m = +
- 30. Classwork ⚫ College Algebra 2e ⚫ 1.5: 4-14 (even); 1.4: 18-36 (even); 1.3: 52-66 (even) ⚫ 1.5 Classwork Check ⚫ Quiz 1.4