7.3 products and factors of polynomials objectives: multiply polynomials, and divide one polynomial...

7.3 Products and Factors 7.3 Products and Factors of Polynomialsof Polynomials

Objectives: Multiply polynomials, and Objectives: Multiply polynomials, and divide one polynomial by another by divide one polynomial by another by

using long division and synthetic using long division and synthetic division.division.

Standard: 2.8.11.S. Analyze properties Standard: 2.8.11.S. Analyze properties and relationships of polynomials.and relationships of polynomials.

Ex 2A. Write f(x) = 2x2 (x2 + 2) (x - 3) as a polynomial in standard form.

(2x4 + 4x2) (x – 3)

2x5 – 6x4 + 4x3 – 12x2

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.Ex 3. Factor each polynomial.

xx33 - 5 - 5xx22 - 6 - 6xx

xx33 + 4 + 4xx22 +2+2xx + 8 + 8

Ex 3. Factor each polynomial.Ex 3. Factor each polynomial. a.a. xx33 – 9– 9xx x (xx (x2 2 –9)–9) x (x + 3) (x – 3)x (x + 3) (x – 3)

b. xb. x33 – – xx22 + 2+ 2xx – 2 – 2 xx22 (x – 1) + 2 (x – 1) (x – 1) + 2 (x – 1) (x(x2 2 + 2) ( x – 1)+ 2) ( x – 1)

c.c. xx33 + 16 + 16xx22 + 64x + 64x x (xx (x2 2 + 16x + 64)+ 16x + 64) x (x + 8) (x + 8)x (x + 8) (x + 8)

c. x3 + 1000

d. x3 – 125

e. x3 + 125

f. x3 – 27

(x + 10) (x2 – 10x + 100)

(x – 5) (x2 + 5x + 25)

(x2 – 5x + 25)(x – 5)

(x – 3) (x2 + 3x + 9)

Factor Theoremx – 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 Use substitution to determine whether x + 3 is a factor of xa factor of x33 – 3x – 3x22 – 6x + 8. – 6x + 8.

f(x) = xf(x) = x33 – 3x – 3x22 – 6x + 8 – 6x + 8

Write x + 3 as x – (-3)Write x + 3 as x – (-3)

Find f(-3)Find f(-3)

f(-3) = (-3)f(-3) = (-3)33 – 3(-3) – 3(-3)22 – 6(-3) + 8 – 6(-3) + 8

= -27 – 27 + 18 + 8= -27 – 27 + 18 + 8

= -28= -28

Since f(-3) does not equal 0; No, its notSince 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.Find the quotient.(x(x33 + 3x + 3x22 – 13x – 15) – 13x – 15) ÷÷ (x (x22 – 2x – 3) – 2x – 3)

* * Given that -3 is a zero of P(Given that -3 is a zero of P(xx) = ) = xx33 - 13 - 13xx – 12, – 12,

use synthetic division to factor use synthetic division to factor xx33 - 13 - 13xx – 12. – 12.

Remainder TheoremIf the polynomial expression that

defines the function of P is divided by x – r, then the remainder is the number P(r).

* * Ex 11. Given P(Ex 11. Given P(xx) = 3) = 3xx33 + 2 + 2xx22 – 3 – 3x x + 5, + 5, find P(3). find P(3).

* * Ex 12. Given P(Ex 12. Given P(xx) = 3) = 3xx33 - 4 - 4xx22 + 9 + 9x x + 5, find P(6) + 5, find P(6) by using both synthetic division and by using both synthetic division and

substitution.substitution.

Writing ActivitiesWriting Activities

Review ofReview of Products and Factors of PolynomialsProducts and Factors of Polynomials

Homework

Integrated Algebra II- Section 7.3 Level A

Academic Algebra II- Section 7.3 Level B