§ 5.5

§ 5.5

Factoring Special Forms

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 5.6

A Strategy for Factoring Polynomials, page 363

1. If there is a common factor, factor out the GCF or factor out a common factor with a negative coefficient.

2. Determine the number of terms in the polynomial and try factoring as follows:

(a) If there are two terms, can the binomial be factored by using one of the following special forms.

Difference of two squares: Sum and Difference of two cubes:

(b) If there are three terms,

If is the trinomial a perfect square trinomial

use one of the adjacent forms:

If the trinomial is not a perfect square trinomial,

If a is equal to 1, use the trial and error

If a is > than 1, use the grouping method

(c) If there are four or more terms, try factoring by grouping.

BABABA 22

///////////////////////////////////////////

222 2 BABABA 222 2 BABABA


The Difference of Two Squares

The Difference of Two SquaresIf A and B are real numbers, variables, or algebraic expressions, then

In words: The difference of the squares of two terms, factors as the product of a sum and a difference of those terms.

. 22 BABABA

P 353



EXAMPLEEXAMPLE

SOLUTIONSOLUTION

.yx 64 925 Factor:

We must express each term as the square of some monomial. Then we use the formula for factoring . 22 BABABA

64 925 yx

2322 35 yx

3232 3535 yxyx

Express as the difference of two squares

Factor using the Difference of Two Squares method



Check Point 1aCheck Point 1a

Factor 2516 2 x

Check Point 1bCheck Point 1b

Factor 46 9100 xy

310310 2323 xyxy

5-4x54 x

310y

P 354

23x



EXAMPLEEXAMPLE

SOLUTIONSOLUTION

.yx 22 66 Factor:

The GCF of the two terms of the polynomial is 6. We begin by factoring out 6.

22 66 yx

226 yx

yxyx 6

Factor the GCF out of both terms

Factor using the Difference of Two Squares method



Check Point 2Check Point 2

Factor 7266 yxy

33 1xy16y xy

P 355

)1(6 62 yxy



EXAMPLEEXAMPLE

SOLUTIONSOLUTION

.x 14 Factor completely:

2224 11 xx

11 22 xx

222 11 xx

Express as the difference of two squares

The factors are the sum and difference of the expressions being squared

The factor is the difference of two squares and can be factored

12 x



1112 xxx The factors of are the sum and difference of the expressions being squared

12 x

CONTINUECONTINUEDD

Thus, . 1111 24 xxxx




Factor 8116 4 x

9494x 22 x

P 355

323294x2 xx


Factoring Completely

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

.xxx 2793 23 Factor completely:

Group terms with common factors

92 x

2793 23 xxx

2793 23 xxx

3932 xxx

93 2 xx

333 xxx

Factor out the common factor from each group

Factor out x + 3 from both terms

Factor as the difference of two squares




Factor 2847 23 xxx

47x 2 x

P 355

227x xx


Factoring Special Forms

Factoring Perfect Square TrinomialsLet A and B be real numbers, variables, or algebraic expressions.

222 2 )1 BABABA

222 2 )2 BABABA

P 356


Factoring Perfect Square Trinomials

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

.yxyx 22 254016 Factor:

We suspect that is a perfect square trinomial because . The middle term can be expressed as twice the product of 4x and -5y.

2222 525 and 416 yyxx

Express in form

Factor

22 254016 yxyx

22 254016 yxyx

22 55424 yyxx

254 yx

22 2 BABA


Factoring Perfect Square Trinomials

Check Point 5aCheck Point 5a

Factor 962 xx

Check Point 5bCheck Point 5b

Factor

23x

P 357

22 254016 yxyx

254 yx Check Point 5bCheck Point 5b

Factor 25204 24 yy

22 52 y


Grouping & Difference of Two Squares

EXAMPLE, use for #65 and #67 (not on test)EXAMPLE, use for #65 and #67 (not on test)

SOLUTIONSOLUTION

.xxx 9624 Factor:

Group as minus a perfect square trinomial to obtain a difference of two squares

Factor the perfect square trinomial

4x

9624 xxx

9624 xxx

24 3 xx

222 3 xx Rewrite as the difference of two squares


Grouping & Difference of Two Squares

Factor the difference of two squares. The factors are the sum and difference of the expressions being squared.

Simplify

33 22 xxxx

33 22 xxxx

CONTINUECONTINUEDD

Thus, . 3396 2224 xxxxxxx




Factor22 2510 yxx


Factor 4422 bba

22 baba

y-5x5 yx

P 357-8


Special Polynomials

In this section we will consider some polynomials that have special formsthat make it easy for us to see how they factor. You may look at a polynomial and say, “Oh, that’s just a difference of squares” or “I think we have a sum of cubes here.” When you have a special polynomial, in particular one that is a difference of two squares, a perfect square polynomial, or a sum or difference of cubes, you will have a factoring formula memorized and will know how to proceed.

That’s why these polynomials are “special”. They may just become our bestfriends among the polynomials.….


The Sum & Difference of Two Cubes

Factoring the Sum & Difference of Two Cubes1) Factoring the Sum of Two Cubes:

Same Signs Opposite Signs

2) Factoring the Difference of Two Cubes:

Same Signs Opposite Signs

2233 BABABABA

2233 BABABABA



EXAMPLEEXAMPLE

SOLUTIONSOLUTION

.yx 6433 Factor:

Rewrite as the Sum of Two Cubes

Factor the Sum of Two Cubes

Simplify

6433 yx 33 4 xy

22 444 xyxyxy

1644 22 xyyxxy

Thus, . 164464 2233 xyyxxyyx



EXAMPLEEXAMPLE

SOLUTIONSOLUTION

.yx 66 64125 Factor:

Rewrite as the Difference of Two Cubes

Factor the Difference of Two Cubes

Simplify

3232 45 yx

22222222 445545 yyxxyx

422422 16202545 yyxxyx

Thus,

66 64125 yx

. 1620254564125 42242266 yyxxyxyx

§ 5.5

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