1. 7.3 Products and Factors of Polynomials Objectives: Multiply polynomials, and divide one polynomial by another by using long division and synthetic division. Standard: 2.8.11.S. Analyze properties and relationships of polynomials.

2. Ex 2A. Write f(x) = 2x 2 (x 2 + 2) (x - 3) as a polynomial in standard form. (2x 4 + 4x 2 ) (x – 3) 2x 5 – 6x 4 + 4x 3 – 12x 2 Ex. 2B Write as a polynomial in standard form.

3. Just as a quadratic expression is factored by writing it as a product of two factors, a polynomial expression of a degree greater than 2 is factored by writing it as a product of more than two factors.

4. Ex 3. Factor each polynomial. x 3 - 5x 2 - 6x x 3 - 5x 2 - 6x x 3 + 4x 2 +2x + 8 x 3 + 4x 2 +2x + 8

5. Ex 3. Factor each polynomial. x 3 – 9x x 3 – 9x x (x 2 –9) x (x 2 –9) x (x + 3) (x – 3) x (x + 3) (x – 3) x 3 – x 2 + 2x – 2 x 3 – x 2 + 2x – 2 x 2 (x – 1) + 2 (x – 1) x 2 (x – 1) + 2 (x – 1) (x 2 + 2) ( x – 1) (x 2 + 2) ( x – 1) x 3 + 16x 2 + 64x x 3 + 16x 2 + 64x x (x 2 + 16x + 64) x (x 2 + 16x + 64) x (x + 8) (x + 8) x (x + 8) (x + 8)

7. c. x 3 + 1000 d. x 3 – 125 e.x 3 + 125 f.x 3 – 27 (x + 10)(x 2 – 10x + 100) (x – 5)(x 2 + 5x + 25) (x 2 – 5x + 25)(x – 5) (x – 3) (x 2 + 3x + 9)

8. Factor Theorem x – r is a factor of the polynomial expression that defines the function P if and only if r is a solution of P(x) = 0, that is, if and only if P(r) = 0. With the Factor Theorem, you can test for linear factors involving integers by using substitution.

10. Use substitution to determine whether x + 3 is a factor of x 3 – 3x 2 – 6x + 8. f(x) = x 3 – 3x 2 – 6x + 8 Write x + 3 as x – (-3) Find f(-3) f(-3) = (-3) 3 – 3(-3) 2 – 6(-3) + 8 = -27 – 27 + 18 + 8 = -27 – 27 + 18 + 8 = -28 = -28 Since f(-3) does not equal 0; No, its not Since f(-3) does not equal 0; No, its not a factor. a factor.

11. DIVIDING POLYNOMIALS BY SYNTHETIC OR LONG DIVISION A polynomial can be divided by a divisor of the form x – r (FIRST POWER) by using long division or a shortened form of long division called synthetic division.

15. * Find the quotient. (x 3 + 3x 2 – 13x – 15) ÷ (x 2 – 2x – 3)

17. * Given that -3 is a zero of P(x) = x 3 - 13x – 12, use synthetic division to factor x 3 - 13x – 12.

18. Remainder Theorem If the polynomial expression that defines the function of P is divided by x – a, then the remainder is the number P(a).

20. * Ex 11. Given P(x) = 3x 3 + 2x 2 – 3x + 5, find P(3).

21. * Ex 12. Given P(x) = 3x 3 - 4x 2 + 9x + 5, find P(6) by using both synthetic division and substitution.

22. Writing Activities

23. Review of Products and Factors of Polynomials

24. Homework Pg. 445-446 #14-98 even