Chapter 6 Section 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Chapter Chapter 66Section Section 44

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley


Special Factoring Techniques

Factor a difference of squares.Factor a perfect square trinomial.Factor a difference of cubes.Factor a sum of cubes.

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44

33

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6.46.46.46.4


By reversing the rules for multiplication of binomials from Section 5.6, we get rules for factoring polynomials in certain forms.

Special Factoring Techniques

Slide 6.4 - 3


Objective 11

Factor a difference of squares.

Slide 6.4 - 4


Factor a difference of squares.The formula for the product of the sum and difference of the

same two terms is

Slide 6.4 - 5

Reversing this rule leads to the following special factoring rule.

2 2.x y x y x y

2 2x y x y x y

The following conditions must be true for a binomial to be a difference of squares:

1. Both terms of the binomial must be squares, such asx2, 9y2, 25, 1, m4.

2. The second terms of the binomials must have different signs (one positive and one negative).

2 2 216 4 4 4m m m m For example, .


EXAMPLE 1

Factor each binomial if possible.Solution:

Factoring Differences of Squares

Slide 6.4 - 6

2 81t 2 2r s

2 16

25x

2 10t 2 36t

9 9t t

r s r s

4 4

5 5x x

prime

prime

After any common factor is removed, a sum of squares cannot be factored.


Factor each difference of squares.

EXAMPLE 2Factoring Differences of Squares

Slide 6.4 - 7

Solution:

249 25x

2 264 81a b

7 5 7 5x x

8 9 8 9a b a b

You should always check a factored form by multiplying.


EXAMPLE 3

Factor completely.

Factoring More Complex Differences of Squares

Slide 6.4 - 8

Solution:250 32r

4 100z

4 81z

22 25 16r

2 210 10z z

2 29 9z z

2 5 4 5 4r r

Factor again when any of the factors is a difference of squares as in the last problem.

Check by multiplying.

2 9 3 3z z z


Objective 22

Factor a perfect square trinomial.

Slide 6.4 - 9


The expressions 144, 4x2, and 81m6 are called perfect squares because

A perfect square trinomial is a trinomial that is the square of a binomial. A necessary condition for a trinomial to be a perfect square is that two of its terms be perfect squares. Even if two of the terms are perfect squares, the trinomial may not be a perfect square trinomial. We can multiply to see that the square of a binomial gives one of the following perfect square trinomials.

Factor a perfect square trinomial.

Slide 6.4 - 10

2144 ,12 224 ,2x x 26 381 9 .m m

22 22x xy y x y

22 22x xy y x y

and


EXAMPLE 4

Solution:

210k

Factoring a Perfect Square Trinomial

Slide 6.4 - 11

Factor k2 + 20k + 100.

2 20 100k k

Check :

2 10 20k k


EXAMPLE 5

Factor each trinomial.

Factoring Perfect Square Trinomials

Slide 6.4 - 12

Solution:2 24 144x x

225 30 9x x

236 20 25a a

212x

25 3x

prime

3 218 84 98x x x 22 9 42 49x x x 22 3 7x x


1. The sign of the second term in the squared binomial is always the same as the sign of the middle term in the trinomial.

Slide 6.4 - 13

Factoring Perfect Square Trinomials

3. Perfect square trinomials can also be factored by using grouping or the FOIL method, although using the method of this section is often easier.

2. The first and last terms of a perfect square trinomial must be positive, because they are squares. For example, the polynomial x2 – 2x – 1 cannot be a perfect square, because the last term is negative.


Objective 33

Slide 6.4 - 14

Factor a difference of cubes.


Factor a difference of cubes.

Just as we factored the difference of squares in Objective 1, we can also factor the difference of cubes by using the following pattern.

This pattern for factoring a difference of cubes should be memorized.

Slide 6.4 - 15

3 3 2 2x y x y x xy y

The polynomial x3 − y3 is not equivalent to (x − y )3, because (x − y )3 can also be written as

but

3x y x y x y x y

2 22x y x xy y

3 3 2 2x y x y x xy y

same sign

positive

opposite sign


Factor each polynomial.

EXAMPLE 6 Factoring Differences of Cubes

Slide 6.4 - 16

Solution:3 216x

327 8x 35 5x

26 6 36x x x

23 2 9 6 4x x x

3 664 125x y 2 2 24 5 16 20 25x y x xy

35 1x 25 1 1x x x

A common error in factoring a difference of cubes, such as x3 − y3 = (x − y)(x2 + xy + y2), is to try to factor x2 + xy + y2. It is easy to confuse this factor with the perfect square trinomial x2 + 2xy + y2. But because there is no 2, it is unusual to be able to further factor an expression of the form x2 + xy +y2.


Objective 44

Slide 6.4 - 17

Factor a sum of cubes.


Factor a sum of cubes.

A sum of squares, such as m2 + 25, cannot be factored by using real numbers, but a sum of cubes can be factored by the following pattern.

Slide 6.4 - 18

Note the similarities in the procedures for factoring a sum of cubes and a difference of cubes.

1. Both are the product of a binomial and a trinomial.2. The binomial factor is found by remembering the “cube root, same

sign, cube root.”3. The trinomial factor is found by considering the binomial factor and

remembering, “square first term, opposite of the product, square last term.”

3 3 2 2x y x y x xy y same sign

positive

opposite sign


The methods of factoring discussed in this section are summarized here.

Slide 6.4 - 19


EXAMPLE 7 Factoring Sums of Cubes

Slide 6.4 - 20

Factor each polynomial.

Solution:

3 64p

3 327 64x y

6 3512a b

24 4 16p p p

2 23 4 9 12 16x y x xy y

2 4 2 28 64 8a b a a b b

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Chapter 6 Section 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley