§ 5.3

§ 5.3

Greatest Common Factors and Factoring by Grouping

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 5.3

Factoring

bybxyaxa 4263 22

Factoring a polynomial means finding an equivalent expression that is a product. For example, when we take the polynomial

bayx 232 2

And write it as

we say that we have factored the polynomial. In factoring, we write asum as a product.


Factoring

Factoring a Monomial from a Polynomial1) Determine the greatest common factor of all terms in the polynomial.

2) Express each term as the product of the GCF and its other factor.

3) Use the distributive property to factor out the GCF.


Factoring

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

344453534 357049 zyxzyxzyx

. 7 333 zyx

. 357049 344453534 zyxzyxzyx Factor:

The GCF is

First, we determine the greatest common factor of the three terms.

Notice that the greatest integer that divides into 49, 70 and 35 (the coefficients of the terms) is 7. The variables raised to the smallest exponents are . and , 333 zyx

P 332


Factoring

xyzyxzyzyxxzzyx 5710777 33323332333

344453534 357049 zyxzyxzyx

Factor out the GCF

Express each term as the product of the GCF and its other factor

CONTINUECONTINUEDD

xyzyxzzyx 51077 22333

P 332


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


Factoring

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

xxx 10505 23

251055 2 xxxxx

. 10505 23 xxx Factor:

The GCF is 5x. Because the leading coefficient, -5, is negative, we factor out a common factor with a negative coefficient. We will factor out the negative of the GCF, or -5x.

Factor out the GCF 2105 2 xxx

Express each term as the product of the GCF and its other factor

P 333

Important to factor the

negative sign


Factoring

Check Point 1Check Point 1

Factor xx 3020 2

Check Point 2aCheck Point 2a

Factor 24 219 xx

10x

733 22 xx

32x10x

23x

P 332-333


Factoring

Check Point 2bCheck Point 2b

Factor 3423 2515 yxyx

Check Point 2cCheck Point 2c

Factor 324354 4816 yxyxyx

235x y

1244 2232 xyyxyx

5xy-35x 23 y

324x y

P 332-333


Factoring


Factor xxx 6102 23

2x-

35x-x2x- 2

P 334

Want a to be positive


Factoring by Grouping

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

yxyxx 7

. 7 yxyxx Factor:

Let’s identify the common binomial factor in each part of the problem.

The GCF, a binomial, is x + y.

The GCF, a binomial, is x + y.

P 334



yxyxx 17

17 xyx

yxyxx 7

We factor out the common binomial factor as follows.

Factor out the GCF

This step, usually omitted, shows each term as the product of the GCF and its other factor, in that order.

CONTINUECONTINUEDD


Factoring

Check Point 4aCheck Point 4a

Factor 4743 xax

Check Point 4bCheck Point 4b

Factor

4-x

7a34 x

P 334

babax 7

ba

1-7xba



Factoring by Grouping1) Group terms that have a common monomial factor. There will usually be two groups. Sometimes the terms must be rearranged.

2) Factor out the common monomial factor from each group.

3) Factor out the remaining common binomial factor (if one exists).

P 334



EXAMPLEEXAMPLE

SOLUTIONSOLUTION

by .bxyaxa 4263 22

yaxa 22 63

Factor:

There is no factor other than 1 common to all terms. However, we can group terms that have a common factor:

Common factor is : Use -2b, rather than 2b, as the common factor: -2bx – 4by = -2b(x + 2y). In this way, the common binomial factor, x + 2y, appears.

bybx 42

yxayaxa 2363 222

+

23a

P 334



bybxyaxa 4263 22

bybxyaxa 4263 22

The voice balloons illustrate that it is sometimes necessary to use a factor with a negative coefficient to obtain a common binomial factor for the two groupings. We now factor the given polynomial as follows:

Group terms with common factors

bayx 232 2

yxbyxa 2223 2

CONTINUECONTINUEDD

Factor out the common factors from the grouped terms

Factor out the GCF

P 334


Factoring


Factor 2054 23 xxx


Factor

5x4 2 x

P 335-6

yxyxx 153204 2

3y-4x5x



EXAMPLEEXAMPLE

Your local electronics store is having an end-of-the-year sale. The price on a large-screen television had been reduced by 30%. Now the sale price is reduced by another 30%. If x is the television’s original price, the sale price can be represented by

(x – 0.3x) – 0.3(x – 0.3x)

(a) Factor out (x – 0.3x) from each term. Then simplify the resulting expression.

(b) Use the simplified expression from part (a) to answer these questions. With a 30% reduction followed by a 30% reduction, is the television selling at 40% of its original price? If not, at what percentage of the original price is it selling?



(a) (x – 0.3x) – 0.3(x – 0.3x)

SOLUTIONSOLUTION

CONTINUECONTINUEDD

= 1(x – 0.3x) – 0.3(x – 0.3x)

= (x – 0.3x)(1 – 0.3)

= (x – 0.3x)(0.7)

= 0.7x – 0.21x

This step, usually omitted, shows each term as the product of the GCF and its other factor, in that order.

Factor out the GCF

Subtract

Distribute

= 0.49x Subtract



(b) With a 30% reduction, followed by another 30% reduction, the expression that represents the reduced price of the television simplifies to 0.49x. Therefore, this series of price reductions effectively gives a new price for the television at 49% its original price, not 40%.

CONTINUECONTINUEDD

§ 5.3

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