holt mcdougal algebra 2 factoring polynomials use the factor theorem to determine factors of a...

Holt McDougal Algebra 2

Factoring Polynomials

Use the Factor Theorem to determine factors of a polynomial.Factor the sum and difference of two cubes.

Objectives

Holt McDougal Algebra 2

Factoring Polynomials

Recall that if a number is divided by any of its factors, the remainder is 0. Likewise, if a polynomial is divided by any of its factors, the remainder is 0. The Remainder Theorem states that if a polynomial is divided by (x – a), the remainder is the value of the function at a. So, if (x – a) is a factor of P(x), then P(a) = 0.

Holt McDougal Algebra 2

Factoring Polynomials

Determine whether the given binomial is a factor of the polynomial P(x).

Example 1: Determining Whether a Linear Binomial is a Factor

A. (x + 1); (x2 – 3x + 1) B. (x + 2); (3x4 + 6x3 – 5x – 10)

Holt McDougal Algebra 2

Factoring Polynomials

Check It Out! Example 1

Determine whether the given binomial is a factor of the polynomial P(x).

a. (x + 2); (4x2 – 2x + 5) b. (3x – 6); (3x4 – 6x3 + 6x2 + 3x – 30)

Holt McDougal Algebra 2

Factoring Polynomials

Zero-Product Property

Example 1:

Holt McDougal Algebra 2

Factoring Polynomials

Zero-Product Property

Example 2:

-10m3 – 2m2

Holt McDougal Algebra 2

Factoring Polynomials

Zero-Product Property

Example 3:

Holt McDougal Algebra 2

Factoring Polynomials

Factoring Trinomials

Example 1:

Holt McDougal Algebra 2

Factoring Polynomials

Factoring trinomials Example 2:

Holt McDougal Algebra 2

Factoring Polynomials

Factoring trinomials Example 3:

Holt McDougal Algebra 2

Factoring Polynomials

Factoring trinomials Example 4:

Holt McDougal Algebra 2

Factoring Polynomials

Factoring trinomials Example 5:

3x2 – 5x + 2

Holt McDougal Algebra 2

Factoring PolynomialsDetermine whether the given polynomial is a factor of the polynomial P(x):

1.(x – 2); P(x) = 5x3 + x2 – 72.(x – 8); P(x) = x5 – 8x4 + 8x – 64Use the zero-product property to solve the equation:3.(t + 4)(t – 4) = 04.(c + 7)(c + 2) = 05.( d + ¾)(d + 5/8) = 06.(x + 5.4)(x – 3) = 07.(5y – 1)(2y + 4) = 08.(7t + 21)(t + 9) = 09.(2n – 7)(5n + 20) = 010. (2x – ½)(2x + ½) = 0

Holt McDougal Algebra 2

Factoring Polynomials

Factoring trinomials Example 6:

6x3 + 24x2 + 24x

Holt McDougal Algebra 2

Factoring Polynomials

Factoring trinomials Example 7:

2t3 – 2t2 – 12t

Holt McDougal Algebra 2

Factoring Polynomials

Factoring trinomials Example 8:

7x2 – 12x + 5

Holt McDougal Algebra 2

Factoring Polynomials

Factoring trinomials Example 9:

Holt McDougal Algebra 2

Factoring Polynomials

Holt McDougal Algebra 2

Factoring Polynomials

Holt McDougal Algebra 2

Factoring Polynomials

Holt McDougal Algebra 2

Factoring Polynomials

Factor: x3 – x2 – 25x + 25.

Example 2: Factoring by Grouping

Holt McDougal Algebra 2

Factoring Polynomials

Example 2 Continued

Check Use the table feature of your calculator to compare the original expression and the factored form.

The table shows that the original function and the factored form have the same function values.

Holt McDougal Algebra 2

Factoring Polynomials

Check It Out! Example 2a

Factor: x3 – 2x2 – 9x + 18.

Holt McDougal Algebra 2

Factoring Polynomials

Check It Out! Example 2a Continued

Check Use the table feature of your calculator to compare the original expression and the factored form.

The table shows that the original function and the factored form have the same function values.

Holt McDougal Algebra 2

Factoring Polynomials

Check It Out! Example 2b

Factor: 2x3 + x2 + 8x + 4.

Holt McDougal Algebra 2

Factoring Polynomials

Just as there is a special rule for factoring the difference of two squares, there are special rules for factoring the sum or difference of two cubes.

Holt McDougal Algebra 2

Factoring Polynomials

Example 3A: Factoring the Sum or Difference of Two Cubes

Factor the expression.

4x4 + 108x

Holt McDougal Algebra 2

Factoring Polynomials

Example 3B: Factoring the Sum or Difference of Two Cubes

Factor the expression.

125d3 – 8

Holt McDougal Algebra 2

Factoring Polynomials

Check It Out! Example 3a

Factor the expression.

8 + z6

Holt McDougal Algebra 2

Factoring Polynomials

Check It Out! Example 3b

Factor the expression.

2x5 – 16x2

Holt McDougal Algebra 2

Factoring Polynomials

Example 4: Geometry ApplicationThe volume of a plastic storage box is modeled by the function V(x) = x3 + 6x2 + 3x – 10. Identify the values of x for which V(x) = 0, then use the graph to factor V(x).

Holt McDougal Algebra 2

Factoring Polynomials

Check It Out! Example 4

The volume of a rectangular prism is modeled by the function V(x) = x3 – 8x2 + 19x – 12, which is graphed below. Identify the values of x for which V(x) = 0, then use the graph to factor V(x).

Holt McDougal Algebra 2

Factoring Polynomials

4. x3 + 3x2 – 28x – 60

Lesson Quiz

2. x + 2; P(x) = x3 + 2x2 – x – 2

1. x – 1; P(x) = 3x2 – 2x + 5

3. x3 + 3x2 – 9x – 27

4. 64p3 – 8q3