In Arithmetic (3)(5) = 15 Factors Product multiply We multiply factors to form a product. factor We factor a number by expressing it as a product of factors. 15 = (3)(5) number Factors The operations of multiplying and factoring are inverses; that is, each operation reverses the other.

In Algebra 4(x + 2)=4x +8 Factors Product expand We expand an expression to form a product. factor We factor a polynomial by expressing it as a product of factors. 4x + 8=4(x + 2) polynomial Factors The operations of expanding and factoring are inverses; that is, each operation reverses the other.

EXAMPLES Factor fully, 2x 3 + 4x 2

Example Factor Fully: -6y + 3y 2 – 3y 3

Example Simplify, then factor: 5x 2 – 3x + 2 – x x + 10 Step 1: Simplify = 5x 2 – x 2 – 3x + 11x = 4x 2 + 8x + 12 Collect like terms Simplify Step 2: Factor: 4x 2 + 8x + 12 = 4(x 2 + 2x + 3)

8x + 4 3x 3 + 9x -12y + 4y 2 - 2y 3 FACTOR COMPLETELY

7x 2 - 4x x - x FACTOR COMPLETELY 6x 3 + 3x 2 - 2x - 2x - x 2 + 7x 2

Let’s go back to Dividing Polynomials by Monomials (Lesson 3-7) There is another way to divide a polynomial by a monomial. Method #2 factorthen divide common factor We factor the polynomial, and then divide the monomial into the common factor. If necessaryfactor the monomial before dividing. If necessary, we factor the monomial before dividing.

Example 5x 2 – 10x 5 = 5(x 2 – 2x) 5 Factored the numerator. GCF = 5 = 5(x 2 – 2x) 5 The 5 in the numerator cancels out the 5 in the denominator. = x 2 – 2x

Example 2 = Factored the numerator. GCF = 3b = The 3b in the numerator cancels out the 3b in the denominator. = 3b 3 – 6b 2 + 9b 6b 3b(b 2 – 2b + 3) 6b 3b(b 2 – 2b + 3) 3b(2) Factored the monomial in the denominator. 3b(b 2 – 2b + 3) 3b(2) Continued next slide

Example 2 = = = (b 2 – 2b + 3b) 2 b 2 – 2b + 3b b(b 2 – 2b + 3b) 3b(2) Divide each term in the numerator by the term in the denominator. b22b22 - b + 3b 2 Remember that you can have a fraction in your answer, just reduce.

12x 4 – 28x 2 + 4x 4x FACTOR COMPLETELY (Compare Methods) METHOD 1 METHOD 2

Class work Finish yesterday’s handout Text questions p.304 #3aceg, 6ace, 7ace, 8ace, 9ace Assignment SEA is due on Wednesday!