ThenThenYou have already factored monomials. (Lesson 9 -2)

NowNow Use the Greatest

Common Factor to factor polynomials.

Solve problems by using the Greatest Common Factor to factor polynomials.

New VocabularyNew Vocabulary

factoringfactored form

Math Online

glencoe.com

Looking

Ahead7

LA24 Looking Ahead to Algebra 1

Why?Why?

A large area of an art classroom is shown at the

right. The total area is represented by A = 3 x 2 + 9x.

a. The teacher wants to create three rectangular work stations in the classroom that are x feet long. Each station has the same width. Write an expression for the width.

b. If x is equal to 8 feet, what are the dimensions and area of the classroom?

Factoring Sometimes, you know the product and are asked to find the factors. This process is called factoring. You can use algebra tiles to model factoring.

EXAMPLE 1 Using Algebra Tiles to Model Factoring

Use algebra tiles to factor 2x + 8.

Step 1 Model the polynomial 2x + 8.

Step 2 Arrange the tiles into a rectangle. The total area of the tiles represents the product. Its length and width represent the factors. The rectangle has a width of 2 and a length of x + 4.

So, 2x + 8 = 2(x + 4).

Guided Practice✓Factor. Use algebra tiles.

1A. 3x + 9 1B. 4x + 10 1C. x 2 + 5x 1D. 3 x 2 + 4x

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A polynomial is in factored form when it is expressed as the product of polynomials. To factor 8 y 2 + 10y, first find the greatest common factor (GCF) of each term.

8 y 2 = 2 · 2 · 2 · y · y Write the prime factorization of 8 y 2 and 10y.

10y = 2 · 5 · y

Circle the common factors.

So, the GCF of 8 y 2 and 10y is 2y.

Write each term as a product of the GCF and its remaining factors. Then use the Distributive Property.

8 y 2 + 10y = 2y(4y) + 2y(5)

= 2y(4y + 5) Distributive Property

Paint/SketchStation

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Using the GCF to Factor Polynomials

Lesson 7 Using the GCF to Factor Polynomials LA25

EXAMPLE 2 Using GCF to Factor Polynomials

Factor 30 x 2 + 12x.

First, find the GCF of 30 x 2 and 12x.

30 x 2 = 2 · 3 · 5 · x · x Write the prime factorization of 30x 2 and 12x.

12x = 2 · 2 · 3 · · x Circle the common factors.

The GCF of 30 x 2 and 12x is 6x. Write each term as a product of the GCF and its remaining factors.

30 x 2 + 12x = 6x(5x) + 6x(2)

= 6x(5x + 2) Distributive Property

Guided Practice✓Factor each polynomial using the GCF.

2A. 15a b 2 - 25abc 2B. 18 x 2 y + 12x y 2 + 6xy

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Solve Problems You can solve some real-world problems by factoring.

EXAMPLE 3

ART Kiyoshi is planning to mat a square painting with a mat that is 6 inches wide. Let x represent the length and the width of the painting. Write an expression in factored form that represents the area of the mat.

Step 1 Find the total area.

A = �w

= (x + 12)(x + 12)

= x 2 + 24x + 144

Step 2 Find the area of the mat alone.

Area of mat = Total Area - Area of painting

= x 2 + 24x + 144 - x 2

= 24x + 144

Step 3 Factor.

24x = 2 · 2 · 2 · 3 · x Write the prime factoriztion of 24x and 144.

144 = 2 · 2 · 2 · 2 · 3 · 3 Circle the common factors. The GCF is 24.

24x + 144 = 24(x) + 24(6)

= 24(x + 6) Distributive Property

So, the area of the mat is 24(x + 6) inches.

Guided Practice✓ 3. GEOMETRY If the area of a rectangle is 16 y 2 + 48y, write the expression in

factored form.

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StudyTip

Greatest Common Factor (GCF) Recall that the GCF is the greatest of the factors common to two or more numbers or terms.

x + 12

x + 12

LA26 Looking Ahead to Algebra 1

✓ Check Your Understanding

Use algebra tiles to factor each binomial.

1. 24y + 18 y 2 2. x 2 + 2x 3. 5 + 5 y 2

Factor each polynomial using the GCF.

4. 3x + 6 5. 2 x 2 + 4x 6. 12 a 2 b + 6a

7. GEOMETRY The diagram represents a walkway that is 2 meters wide surrounding a rectangular garden. Write an expression in factored form that represents the area of the walkway.

Practice and Problem Solving

Factor each polynomial. If the polynomial cannot be factored, write cannot be factored.

8. 9x + 15 9. 7 a 2 b 2 + 3a b 3 10. 36mn - 11m n 2

11. 14m n 2 - 2mn 12. 24xy + 18x y 2 - 3y 13. 12axy - 14ay + 20ax

14. The figure at the right shows a walkway built around a statue with a square base.

a. If the walkway is 3 meters wide, write an expression in factored form that represents the area of the

walkway.

b. If the base of the statue is 3 meters wide, find the

area of the walkway.

15. MARINE BIOLOGY In a pool at an aquarium, a dolphin jumps out of the water traveling at 20 feet per second. Its height h, in feet, above the water after t seconds is given by the formula h = 20t - 16 t 2 . Write an expression in factored form that represents the height.

H.O.T. Problems Use Higher-Order Thinking Skills

16. CHALLENGE The area of a circle is found using the formula A = π r 2 , where r is the radius of the circle. Using π, write an expression in factored form that represents the area of the shaded region at the right.

17. REASONING Explain what it means to factor a polynomial.

18. OPEN ENDED Write a polynomial that can be factored. Write another polynomial that cannot be factored.

19. WRITING IN MATH Write a few sentences explaining how the Distributive Property is used to factor polynomials. Include at least two examples.

Example 1p. LA24

Example 1p. LA24

Example 2p. LA25

Example 2p. LA25

Example 3p. LA25

Example 3p. LA25 2x

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Examples 1 and 2pp. LA24, LA25

Example 3p. LA25

Example 3p. LA25

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