§ 5.6

A General Factoring Strategy

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 5.6

Factoring a Polynomial

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

.yyx 54 232 Factor:

1) If there is a common factor, factor out the GCF. Because 2y is common to both terms, we factor it out.

Factor out the GCF 4454 162232 yxyyyx

2) Determine the number of terms and factor accordingly. The factor has two terms. This binomial can be expressed as , so it can be factored as the difference of two squares.

4416 yx 22224 yx

Blitzer, Intermediate Algebra, 5e – Slide #4 Section 5.6

Factoring a Polynomial

3) Check to see if factors can be factored further. We note that is the difference of two squares, , so we continue factoring.

Rewrite as the difference of two squares

CONTINUECONTINUEDD

Factor

4454 162232 yxyyyx

222242 yxy

2222 442 yxyxy

224 yx 222 yx

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

Factoring a Polynomial

The previous factorization

CONTINUECONTINUEDD

Rewrite last factor as the difference of two squares

2222 442 yxyxy

54 232 yyx

2222 242 yxyxy

yxyxyxy 2242 22 Factor

Blitzer, Intermediate Algebra, 5e – Slide #6 Section 5.6

Factoring a Polynomial

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

.yxyx 22 550125 Factor:

1) If there is a common factor, factor out the GCF. Because 5 is common to both terms, we factor it out.

Factor out the GCF

2) Determine the number of terms and factor accordingly. The factor has three terms and is a perfect square trinomial. We factor using .

2222 10255550125 yxyxyxyx

22 1025 yxyx 222 2 BABABA

Blitzer, Intermediate Algebra, 5e – Slide #7 Section 5.6

Factoring a Polynomial

Factor out the GCF

22 550125 yxyx

22 5255 yyxx 222 2 BABABA

3) Check to see if factors can be factored further. In this case, they cannot, so we have factored completely.

CONTINUECONTINUEDD

22 10255 yxyx

255 yx

Rewrite the part in parentheses in

Factor

form

DONE

Blitzer, Intermediate Algebra, 5e – Slide #9 Section 5.6

Factoring a Polynomial

We have looked at factoring out a common factor, factoring by grouping,factoring a difference of squares, factoring general trinomials using trialand error, factoring a sum or difference of cubes, and factoring other special forms.

It is important when you wish to factor a polynomial to know where to start. You should always look first to see if there is a common factor that you can factor out. Do that first. Then consider the number of terms in thepolynomial.

Strategies for factoring a polynomial based on the number ofterms in the polynomial follow.

Blitzer, Intermediate Algebra, 5e – Slide #10 Section 5.6

Factoring a Polynomial

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

.x 324 3 Factor:

1) If there is a common factor, factor out the GCF. Because 3 is common to both terms, we factor it out.

Factor out the GCF 183324 33 xx

2) Determine the number of terms and factor accordingly. The factor has two terms. This binomial can be expressed as , so it can be factored as the difference of two cubes.

18 3 x 33 12 x

Blitzer, Intermediate Algebra, 5e – Slide #11 Section 5.6

Factoring a Polynomial

3) Check to see if factors can be factored further. In this case, they cannot, so we have factored completely.

Rewrite as the difference of two cubes

Simplify

33 123 x

183324 33 xx

CONTINUECONTINUEDD

22 1122123 xxx

124123 2 xxx

Factor

Factoring Strategy

1. Is there a common factor? If so, factor out the GCF.2. How many terms?

If two termsa. The difference of two squaresb. The sum of two cubesc. The difference of two cubes

• If three termsa. A perfect square trinomialb. If a is equal to 1, use the trial-and-check.c. If a is > than 1, use the grouping method.

• If it has four or more terms, factor by grouping

Blitzer, Intermediate Algebra, 5e – Slide #13 Section 5.6

A Strategy for Factoring Polynomials

A Strategy for Factoring a Polynomial (b) If there are three terms, is the trinomial a perfect square trinomial? If

so, factor by using one of the following forms:

If the trinomial is not a perfect square trinomial, try factoring by trial and error or grouping.

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

3) Check to see if any factors with more than one term in the factored polynomial can be factored further. If so, factor completely.

Remember to check the factored form by multiplying or by using the TABLE or GRAPH feature of a graphing utility.

CONTINUECONTINUEDD

222 2 BABABA

222 2 BABABA