1. FACTORING Objective: To factor polynomials using a variety of methods

2. Definitions:  Product:  The result of a multiplication problem  Factor:  The numbers and/or terms that are being multiplied  Factoring:  The process of breaking a number or polynomial into its smallest factors

3. Types of Factoring  Greatest Common Factor  Difference of Squares  Perfect Square Trinomial  Factoring a trinomial into two binomials (using guess and check)  Sum/Difference of Cubes

4. Greatest Common Factor  The GCF is the largest monomial that will divide evenly into EVERY term of the polynomial  You should always look for a GCF first when doing any factoring problem  Ex.  In 20x 4 + 35x 2 the GCF is 5x 2  In 32a 3 b 5 – 24a 2 b 7 – 40ab 8 the GCF is 8ab 5  When checking for a GCF, put the polynomial in standard form. If the leading coefficient is negative, then the GCF should be negative.

5. Greatest Common Factor  Once you have determined the GCF, you will divide it out of each term. The result should be written as  GCF(polynomial result with GCF factored out)  Example:  20x 4 + 35x 2 GCF = 5x 2    20x 4 + 35x 2 = 5x 2 (4x 2 +7)

6. Greatest Common Factor  Example:  6a 3 b 5 – 9a 4 b 6 + 15ab 8 – 18a 2 b 7  Put in standard form:-9a 4 b 6 + 6a 3 b 5 – 18a 2 b 7 + 15ab 8  GCF:-3ab 5  Factored form:-3ab 5 (3a 3 b – 2a 2 + 6ab 2 – 5b 3 )  Example:  12m 4 – 16m 3 + 24m 2  GCF:4m 2  Factored form:4m 2 (3m 2 – 4m + 6)

7. Factoring a Difference of Perfect Squares  Multiply:(2x – 3)(2x + 3)  4x 2 + 6x – 6x – 9  4x 2 – 9  4x 2 – 9 is known as a Difference of Perfect Squares  A Difference of Perfect Squares is a BINOMIAL where both terms are perfect squares and the terms are subtracted.  A Difference of Perfect Squares a 2 – b 2 is factored into two binomials:  (a + b)(a – b)

8. Factoring a Difference of Perfect Squares  Example:  16m 2 – 25  (4m + 5)(4m – 5)  Example:  18a 3 – 98a  GCF:2a  Factor:2a(9a 2 – 49)  Final factored form:2a(3a + 7)(3a – 7)  This is a difference of squares

9. Factoring a Perfect Square Trinomial  Multiply:(2x + 3) 2  (2x + 3)(2x + 3)  4x 2 + 6x + 6x + 9  4x 2 + 12x + 9  A Perfect Square Trinomial is a TRINOMIAL where the first and last terms are perfect squares and the middle term is twice the product of the square roots of the first and last terms (in the example above the square root of the first term is 2x, the square root of the last term is 3, and the middle term is equal to 2  2x  3 or 12x).  The general form of a Perfect Square Trinomial is either a 2 + 2ab + b 2 or a 2 – 2ab + b 2 and factors to either (a + b) 2 or (a – b) 2

10. Factoring a Perfect Square Trinomial  Factor:x 2 + 10x + 25  The first and last terms are perfect squares, and the middle term is double the product of the square roots of the first and last terms.  Factored form:(x + 5) 2  Factor:9x 2 – 24x + 16  The first and last terms are perfect squares (3x and 4) and the middle term is double their product (2  12x)  Factored form:(3x – 4) 2  Factor:25x 2 – 30x – 9  The first and last terms are perfect squares, but the last term is negative so this IS NOT a Perfect Square Trinomial and cannot be factored using the rules for a Perfect Square Trinomial

11. Factoring a General Trinomial  Multiply:(x + 3)(x – 5)  x 2 – 5x + 3x – 15  x 2 – 2x – 15  (x + 3) and (x – 5) are the factors of x 2 - 2x – 15  When factoring a trinomial, first check to see if there is a GCF or if it is a Perfect Square Trinomial. If not, the easiest method is to guess and check to find the factors.

12. Factoring a General Trinomial  Example:Factorx 2 – 4x + 3  Because the leading coefficient is 1, we can look for two numbers whose product is 3 and whose sum is -4. Those two numbers are -3 and -1  The two binomials whose product is x 2 – 4x + 3 (factors) are (x – 3) and (x – 1)  The factored form is (x – 3)(x – 1)  Example:Factor x 2 + 6x – 16  Because the leading coefficient is 1, we can look for two numbers whose product is -16 and whose sum is 6. Those two numbers are 8 and -2.  The factored form is (x + 8)(x – 2)

13. Factoring a General Trinomial  Example:Factor2x 2 + 7x – 15  Because the leading coefficient is not 1, this is a little more difficult to factor.  The first terms of each binomial factor must multiply to be 2x 2, and the second terms of each binomial factor must multiply to be -15. Guess and check different combinations until you find one that works.  Guess #1:(2x – 5)(x + 3) Check:2x 2 + 6x – 5x – 15 = 2x 2 + x – 15  Guess #2:(2x + 3)(x – 5) Check:2x 2 – 10x + 3x – 15 = 2x 2 – 7x – 15  Guess #3:(2x – 3)(x + 5) Check:2x 2 + 10x – 3x – 15 = 2x 2 + 7x - 15  This is not correct!  YEA!!! We have found the correct factors!

14. Factoring a General Trinomial  Example:Factor6x 2 + 11x - 10  The first terms in each binomial factor must multiply to be 6x 2 and the last terms in each binomial factor must multiply to be -10. Guess and check until you find the combination that yields the correct middle term.  (2x + 5)(3x – 2) Check:6x 2 – 4x + 15x – 10 = 6x 2 + 11x – 10  Example:Factor10x 3 + 35x 2 + 15x  There is a GCF of 5x  5x(2x 2 + 7x + 3)  Guess and check until you find the right binomials to factor the trinomial  2x 2 + 7x + 3 factors to be (2x + 1)(x + 3)  Final factored form:5x(2x + 1)(x + 3)

15. Factoring a Sum or Difference of Cubes  A Sum or Difference of Cubes is a BINOMIAL where each term is a perfect cube.  The Sum of Cubes a 3 + b 3 factors into a binomial multiplied by a trinomial with the pattern:  (a + b)(a 2 – ab + b 2 )  The Difference of Cubes a 3 – b 3 factors into a binomial multiplied by a trinomial with the pattern:  (a – b)(a 2 + ab + b 2 )

16. Factoring a Sum or Difference of Cubes  Example:Factor27x 3 + 125  This is a sum of cubes (3x and 5)  The factors are (3x + 5)(9x 2 – 15x + 25)  Example:Factor16x 4 y – 54xy 4  There is a GCF of 2xy  Factoring out the GCF yields 2xy(8x 3 – 27y 3 )  The final factored form is 2xy(2x – 3y)(4x 2 + 6xy + 9y 2 )  This is a difference of cubes!