§ 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
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 5.5
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
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 5.5
The Difference of Two Squares
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
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 5.2
The Difference of Two Squares
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
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 5.5
The Difference of Two Squares
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
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 5.2
The Difference of Two Squares
Check Point 2Check Point 2
Factor 7266 yxy
33 1xy16y xy
P 355
)1(6 62 yxy
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 5.5
The Difference of Two Squares
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
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 5.5
The Difference of Two Squares
1112 xxx The factors of are the sum and difference of the expressions being squared
12 x
CONTINUECONTINUEDD
Thus, . 1111 24 xxxx
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 5.2
The Difference of Two Squares
Check Point 3Check Point 3
Factor 8116 4 x
9494x 22 x
P 355
323294x2 xx
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 5.5
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
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 5.2
The Difference of Two Squares
Check Point 4Check Point 4
Factor 2847 23 xxx
47x 2 x
P 355
227x xx
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 5.5
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
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 5.5
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
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 5.2
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
Blitzer, Intermediate Algebra, 5e – Slide #16 Section 5.5
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
Blitzer, Intermediate Algebra, 5e – Slide #17 Section 5.5
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
DONE
Blitzer, Intermediate Algebra, 5e – Slide #19 Section 5.2
The Difference of Two Squares
Check Point 6Check Point 6
Factor22 2510 yxx
Check Point 7Check Point 7
Factor 4422 bba
22 baba
y-5x5 yx
P 357-8
Blitzer, Intermediate Algebra, 5e – Slide #20 Section 5.5
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.….
Blitzer, Intermediate Algebra, 5e – Slide #21 Section 5.5
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
Blitzer, Intermediate Algebra, 5e – Slide #22 Section 5.5
The Sum & Difference of Two Cubes
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
Blitzer, Intermediate Algebra, 5e – Slide #23 Section 5.5
The Sum & Difference of Two Cubes
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
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