5.5

Factoring Special Forms

Blitzer, Algebra for College Students, 6e – Slide #2 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, Algebra for College Students, 6e – 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

Blitzer, Algebra for College Students, 6e – 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, Algebra for College Students, 6e – Slide #5 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, Algebra for College Students, 6e – Slide #6 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, Algebra for College Students, 6e – Slide #7 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, Algebra for College Students, 6e – Slide #8 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, Algebra for College Students, 6e – Slide #9 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, Algebra for College Students, 6e – Slide #10 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, Algebra for College Students, 6e – Slide #11 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

5.5 Assignment

p.359 (2-18 even, 24-36 even, 76-84 even)