7.3 Products and Factors of Polynomials Objectives: Multiply polynomials, and divide one polynomial by another by using long division and synthetic division.

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Presentation transcript:

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

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.

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.

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

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 x x x x x x (x x + 64) x (x x + 64) x (x + 8) (x + 8) x (x + 8) (x + 8)

c. x d. x 3 – 125 e.x 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)

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.

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 – = -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.

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.

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

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

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).

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

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

Writing Activities

Review of Products and Factors of Polynomials

Homework Pg #14-98 even