Copyright © Cengage Learning. All rights reserved.

Factoring Polynomials andSolving Equations by Factoring 5

Copyright © Cengage Learning. All rights reserved.

Section 5.25.2

Factoring the Difference of Two Squares

3

Objectives

Factor the difference of two squares.

Completely factor a polynomial.

11

22

4

Factoring the Difference of Two Squares

Whenever we multiply binomial conjugates, binomials of the form (x + y) and (x – y), we obtain a binomial of the form x2 – y2.

(x + y)(x – y) = x2 – xy + xy – y2

= x2 – y2

In this section, we will show how to reverse the multiplication process and factor binomials such as x2 – y2 into binomial conjugates.

5

Factor the difference of two squares

1.

6

Factor the difference of two squares

The binomial x2 – y2 is called the difference of two squares, because x2 is the square of x and y2 is the square of y.

The difference of the squares of two quantities always factors into binomial conjugates.

Factoring the Difference of Two Squares

x2 – y2 = (x + y)(x – y)

7

Factor the difference of two squares

To factor x2 – 9, we note that it can be written in the form x2 – 32.

x2 – 32 = (x + 3)(x – 3)

We can check by verifying that (x + 3)(x – 3) = x2 – 9.

To factor the difference of two squares, it is helpful to know the integers that are perfect squares. The number 400, for example, is a perfect square, because 202 = 400.

8

Factor the difference of two squares

The integer squares less than 400 are

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361

Expressions containing variables such as x4y2 are also perfect squares, because they can be written as the square of a quantity:

x4y2 = (x2y)2

9

Example

Factor: 25x2 – 49

Solution:We can write 25x2 – 49 in the form (5x)2 – 72.

25x2 – 49 = (5x)2 – 72

= (5x + 7)(5x – 7)

We can check by multiplying (5x + 7) and (5x – 7).

(5x + 7)(5x – 7) = 25x2 – 35x + 35x – 49

= 25x2 – 49

10

Completely factor a polynomial2.

11

Completely factor a polynomial

We often must factor out a greatest common factor before factoring the difference of two squares.

To factor 8x2 – 32, for example, we factor out the GCF of 8 and then factor the resulting difference of two squares.

8x2 – 32 = 8(x2 – 4)

= 8(x2 – 22)

= 8(x + 2)(x – 2)

Factor out 8, the GCF.

Write 4 as 22.

Factor the difference of two squares.

12

Completely factor a polynomial

We can check by multiplication:

8(x + 2)(x – 2) = 8(x2 – 4)

= 8x2 – 32

13

Example

Factor completely: 2a2x3y – 8b2xy.

Solution:We factor out the GCF of 2xy and then factor the resulting difference of two squares.

2a2x3y – 8b2xy

= 2xy a2x2 – 2xy 4b2

= 2xy(a2x2 – 4b2)

The GCF is 2xy.

Factor out 2xy.

14

Example – Solution

= 2xy[(ax)2 – (2b)2]

= 2xy(ax + 2b)(ax – 2b)

Check by multiplication.

Write a2x2 as (ax)2 and 4b2 as (2b)2.

Factor the difference of two squares.

cont’d

15

Completely factor a polynomial

Sometimes we must factor a difference of two squares more than once to completely factor a polynomial.

For example, the binomial 625a4 – 81b4 can be written in the form (25a2)2 – (9b2)2, which factor as

625a4 – 81b4 = (25a2)2 – (9b2)2

= (25a2 + 9b2)(25a2 – 9b2)

16

Completely factor a polynomial

Since the factor 25a2 – 9b2 can be written in the form (5a)2 – (3b)2, it is the difference of two squares and can be factored as (5a + 3b)(5a – 3b).

Thus,

625a4 – 81b4 = (25a2 + 9b2)(5a + 3b)(5a – 3b)

17

Completely factor a polynomial

The binomial 25a2 + 9b2 is the sum of two squares, because it can be written in the form (5a)2 + (3b)2.

If we are limited to rational coefficients, binomials that are the sum of two squares cannot be factored unless they contain a GCF.

Polynomials that do not factor are called prime polynomials.

18

Your Turn

Factor completely

1.m4 – 16n4

o m4 – 24n4

m4 – (2n)4

(m2 + (2n)2)(m2 – (2n)2)(m2 + 4n2)(m + 2n)(m – 2n)

•a5 – ab4

o a(a4 – b4)a(a2 + b2)(a2 – b2)a(a2 + b2)(a + b)(a – b)

19

Your TurnFactor completely.

3.2x8y2 – 32y6

3. 2y2(x8 – 16y4)2y2((x4)2 – (4y2)2)2y2(x4 + 4y2)(x4 – 4y2)2y2(x4 + 4y2)(x2 + 2y)(x2 – 2y)

4.x8y8 – 1

3. (x4y4)2 – 1(x4y4 + 1)(x4y4 – 1)(x4y4 + 1)(x2y2 + 1)(x2y2 – 1)(x4y4 + 1)(x2y2 + 1)(xy + 1)(xy – 1)

20

Your Turn5. 2a3b – 242ab

o 2ab(a2 – 121)2ab(a2 – 112)2ab(a + 11)(a – 11)

6. 81r4 – 256s4

5. 92r4 – 162s4

(9r2)2 – (16s2)2

(9r2 + 16s2)(9r2 – 16s2)(9r2 + 16s2)(9r2 – 16s2)(9r2 + 16s2)(3r + 4s)(3r – 4s)