Factor the difference of two squares.

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# Factor the difference of two squares. - PowerPoint PPT Presentation

Objectives. Factor the difference of two squares. 4 x 2 – 9. 2 x 2 x 3 3. . . The difference of two squares has the form a 2 – b 2 . The difference of two squares can be written as the product ( a + b )( a – b ). You can use this pattern to factor some polynomials.

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Objectives

Factor the difference of two squares.

4x2–9

2x 2x3 3

The difference of two squares has the form a2–b2. The difference of two squares can be written as the product (a + b)(a – b). You can use this pattern to factor some polynomials.

A polynomial is a difference of two squares if:

• There are two terms, one subtracted from the other.
• Both terms are perfect squares.

Recognize a difference of two squares:

the coefficients of variable terms are perfect squares

powers on variable terms are even,

and constants are perfect squares.

3p2– 9q4

3q2 3q2

Example 3A: Recognizing and Factoring the Difference of Two Squares

Determine whether each binomial is a difference of two squares. If so, factor. If not, explain.

3p2– 9q4

3p2 is not a perfect square.

3p2– 9q4 is not the difference of two squares because 3p2 is not a perfect square.

100x2– 4y2

10x 10x

2y 2y

Example 3B: Recognizing and Factoring the Difference of Two Squares

Determine whether each binomial is a difference of two squares. If so, factor. If not, explain.

100x2– 4y2

The polynomial is a difference of two squares.

(10x)2– (2y)2

a = 10x, b = 2y

(10x + 2y)(10x– 2y)

Write the polynomial as (a + b)(a – b).

100x2– 4y2 = (10x + 2y)(10x– 2y)

x2 x2

5y3 5y3

Example 3C: Recognizing and Factoring the Difference of Two Squares

Determine whether each binomial is a difference of two squares. If so, factor. If not, explain.

x4– 25y6

x4– 25y6

The polynomial is a difference of two squares.

(x2)2– (5y3)2

a = x2, b = 5y3

Write the polynomial as (a + b)(a – b).

(x2 + 5y3)(x2– 5y3)

x4– 25y6 = (x2 + 5y3)(x2– 5y3)

1 1

2x 2x

Check It Out! Example 3a

Determine whether each binomial is a difference of two squares. If so, factor. If not, explain.

1 – 4x2

1 – 4x2

The polynomial is a difference of two squares.

(1) – (2x)2

a = 1, b = 2x

(1 + 2x)(1 – 2x)

Write the polynomial as (a + b)(a – b).

1 – 4x2 = (1+ 2x)(1 – 2x)

p4 p4

7q3 7q3

Check It Out! Example 3b

Determine whether each binomial is a difference of two squares. If so, factor. If not, explain.

p8– 49q6

p8– 49q6

The polynomial is a difference of two squares.

(p4)2– (7q3)2

a = p4, b = 7q3

(p4 + 7q3)(p4– 7q3)

Write the polynomial as

(a + b)(a – b).

p8– 49q6 = (p4 + 7q3)(p4– 7q3)

4x 4x

Check It Out! Example 3c

Determine whether each binomial is a difference of two squares. If so, factor. If not, explain.

16x2– 4y5

16x2– 4y5

4y5 is not a perfect square.

16x2– 4y5 is not the difference of two squares because 4y5 is not a perfect square.

Lesson Quiz: Part II

Determine whether the binomial is a difference of two squares. If so, factor. If not, explain.

5. 9x2 – 144y4

6. 30x2 – 64y2

7. 121x2 – 4y8

(3x + 12y2)(3x– 12y2)

Not a difference of two squares; 30x2 is not a perfect square

(11x + 2y4)(11x – 2y4)