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Choose an appropriate method for factoring a polynomial. - PowerPoint PPT Presentation


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Objectives. Choose an appropriate method for factoring a polynomial. Combine methods for factoring a polynomial. Recall that a polynomial is in its fully factored form when it is written as a product that cannot be factored further.

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slide1

Objectives

Choose an appropriate method for factoring a polynomial.

Combine methods for factoring a polynomial.

slide2

Recall that a polynomial is in its fully factored form when it is written as a product that cannot be factored further.

slide3

Example 1: Determining Whether a Polynomial is Completely Factored

Tell whether each expression is completely factored. If not, factor it.

A. 3x2(6x– 4)

3x2(6x–4)

6x – 4 can be further factored.

6x2(3x– 2)

Factor out 2, the GCF of 6x and – 4.

6x2(3x– 2) is completely factored.

B. (x2 + 1)(x– 5)

(x2 + 1)(x– 5)

Neither x2 +1 nor x – 5 can be factored further.

(x2 + 1)(x– 5) is completely factored.

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Check It Out! Example 1

Tell whether the polynomial is completely factored. If not, factor it.

A. 5x2(x– 1)

Neither 5x2 nor x – 1 can be factored further.

5x2(x– 1)

5x2(x– 1) is completely factored.

B. (4x + 4)(x + 1)

(4x + 4)(x + 1)

4x + 4 can be further factored.

4(x +1)(x + 1)

Factor out 4, the GCF of 4x and 4.

4(x + 1)2 is completely factored.

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To factor a polynomial completely, you may need to use more than one factoring method. Use the steps below to factor a polynomial completely.

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Example 2A: Factoring by GCF and Recognizing Patterns

Factor 10x2 + 48x + 32 completely. Check your answer.

10x2 + 48x + 32

Factor out the GCF.

2(5x2 + 24x + 16)

2(5x + 4)(x + 4)

Factor remaining trinomial.

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Example 2B: Factoring by GCF and Recognizing Patterns

Factor 8x6y2– 18x2y2 completely. Check your answer.

8x6y2– 18x2y2

Factor out the GCF. 4x4 – 9is a perfect-square trinomial of the form a2 – b2.

2x2y2(4x4– 9)

2x2y2(2x2– 3)(2x2 + 3)

a = 2x, b = 3

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4x3 + 16x2 + 16x

4x(x2 + 4x + 4)

Check It Out! Example 2a

Factor each polynomial completely. Check your answer.

4x3 + 16x2 + 16x

4x(x + 2)2

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Check It Out! Example 2b

Factor each polynomial completely. Check your answer.

2x2y– 2y3

2x2y– 2y3

2y(x2–y2)

2y(x + y)(x–y)

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( x + )( x + )

Factors of 9 Factors of –2Outer+Inner

1 and –2

1(–2) + 9(1) = 7

1 and 9

1 and –2

3(–2) + 3(1) = –3

3 and 3

–1 and 2

3(2) + 3(–1) = 3

3 and 3

(3x– 1)(3x + 2)

Example 3A: Factoring by Multiple Methods

Factor each polynomial completely.

9x2 + 3x– 2

The GCF is 1 and there is no pattern.

9x2 + 3x– 2

a = 9 and c = –2;

Outer + Inner = 3

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(b + )(b + )

Factors of 4Sum

1 and 4 5

2 and 2 4

Example 3B: Factoring by Multiple Methods

Factor each polynomial completely.

12b3 + 48b2 + 48b

12b(b2 + 4b + 4)

12b(b + 2)(b + 2)

12b(b + 2)2

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(y + )(y + )

Factors of –18Sum

–1 and 18 17

–2 and 9 7

–3 and 6 3

Example 3C: Factoring by Multiple Methods

Factor each polynomial completely.

4y2 + 12y– 72

4(y2 + 3y– 18)

4(y –3)(y + 6)

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Example 3D: Factoring by Multiple Methods.

Factor each polynomial completely.

(x4–x2)

Factor out the GCF.

x2(x2– 1)

x2 – 1is a difference of two squares.

x2(x + 1)(x– 1)

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( x + )( x + )

Factors of 3 Factors of 4Outer+Inner

1 and 4

3(4) + 1(1) = 13

3 and 1

2 and 2

3(2) + 1(2) = 8

3 and 1

4 and 1

3(1) + 1(4) = 7

3 and 1

(3x + 4)(x + 1)

Check It Out! Example 3a

Factor each polynomial completely.

3x2 + 7x + 4

3x2 + 7x + 4

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(p + )(p + )

Factors of – 6 Sum

– 1 and 6 5

Check It Out! Example 3b

Factor each polynomial completely.

2p5 + 10p4– 12p3

2p3(p2 + 5p– 6)

2p3(p + 6)(p– 1)

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( q + )( q + )

Factors of 3 Factors of 8Outer+Inner

1 and 8

3(8) + 1(1) = 25

3 and 1

2 and 4

3(4) + 1(2) = 14

3 and 1

4 and 2

3(2) + 1(4) = 10

3 and 1

3q4(3q + 4)(q + 2)

Check It Out! Example 3c

Factor each polynomial completely.

9q6 + 30q5 + 24q4

3q4(3q2 + 10q + 8)

slide17

Check It Out! Example 3d

Factor each polynomial completely.

2x4 + 18

2(x4 + 9)

Factor out the GFC.

x4 + 9 is the sum of squares and that is not factorable.

2(x4 + 9) is completely factored.

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Lesson Quiz

Tell whether the polynomial is completely factored. If not, factor it.

1. (x + 3)(5x + 10) 2. 3x2(x2 + 9)

no; 5(x+ 3)(x + 2)

completely factored

Factor each polynomial completely. Check your answer.

3.x3 + 4x2 + 3x + 12 4. 4x2 + 16x – 48

4(x + 6)(x– 2)

(x + 4)(x2 + 3)

5. 18x2– 3x– 3

6. 18x2– 50y2

3(3x + 1)(2x– 1)

2(3x + 5y)(3x– 5y)

7. 5x –20x3 + 7 – 28x2

(1 + 2x)(1 – 2x)(5x + 7)