Chapter Twelve — Factoring Polynomials

The Humongous Book of Algebra Problems

267

Factor the binomial x

2

– 2 out of the expression.

(x

2

– 2)(4x

3

– 5)

Therefore, the factored form of 4x

5

– 8x

3

– 5x

2

+ 10 is (x

2

– 2)(4x

3

– 5).

12.25 Factor the expression: 96x

4

+ 36x

3

– 160x – 60.

Before you factor by grouping, note that all of the terms share the

common factor 4. Factor it out of the expression.

4(24x

4

+ 9x

3

– 40x – 15)

The ﬁrst two terms of the quartic polynomial share common factor 3x

3

;

the last two terms share common factor –5.

Factor the common binomial out of the expression.

Common Factor Patterns

Difference of perfect squares/cubes, sum of perfect cubes

12.26 Factor the expression: a

2

– b

2

.

The expression a

2

– b

2

is a difference of perfect squares. In other words, the

expression consists of one squared quantity (b

2

) subtracted from another (a

2

).

All differences of perfect squares are factored using the formula

(a

2

– b

2

) = (a + b)(a – b).

12.27 Factor the expression: w

2

– 36.

Notice that w

2

and 36 are perfect squares, as each is equal to some quantity

times itself: w · w = w

2

and 6 · 6 = 36.

w

2

– 36 = w

2

– 6

2

According to Problem 12.26, a polynomial of the form a

2

– b

2

has factors

(a + b)(a – b). Here, a = w and b = 6.

In Problem

12.23, fac-

toring out a 1

didn’t change

the binomial. When

you factor out a

–1 in this problem, it

changes every term

in the parentheses

into its opposite.

“Quartic”

means the highest

power of x in the

polynomial is 4.

Leave

this 4 (that

got factored out

in the previous step)

in front of the

expression.

A perfect

square is a

number you get by

multiplying something

times itself. That

means 4 is a perfect

square because 2 times

itself equals 4:

2

˙

2 = 2

2

= 4.

A DIFFERENCE of

perfect squares just

means subtraction

is involved.

Chapter Twelve — Factoring Polynomials

The Humongous Book of Algebra Problems

268

Therefore, the factored form of w

2

– 36 is (w + 6)(w – 6).

12.28 Factor the expression: 4x

2

– 81.

This binomial is the difference of perfect squares: (2x)(2x) = 4x

2

and

(9)(9) = 81. Therefore, it should be factored using the formula

a

2

– b

2

= (a + b)(a – b). If a

2

– b

2

= 4x

2

– 81, then a = 2x and b = 9.

12.29 Factor the expression: 98x

2

– 200.

Both terms are divisible by 2, so factor that value out of the expression.

2(49x

2

– 100)

The resulting binomial, 49x

2

– 100, is a difference of perfect squares: (7x)

2

=

49x

2

and 10

2

= 100.

2(7x + 10)(7x – 10)

12.30 Factor the expression: 4x

4

– 64.

The terms of this expression have a greatest common factor of 4, so factor 4 out

of the polynomial.

4(x

4

– 16)

The binomial x

4

– 16 is a difference of perfect squares: (x

2

)

2

= x

4

and 4

2

= 16.

4(x

2

+ 4)(x

2

– 4)

The expression is not yet fully factored, because x

2

– 4 is, itself, a difference of

perfect squares: (x

2

– 4) = (x + 2)(x – 2). Substitute the factored form of x

2

– 4

into the polynomial.

4x

4

– 64 = 4(x

2

+ 4)(x + 2)(x – 2)

The answer

(w – 6)(w + 6) is also

correct. The order of

the factors doesn’t

matter.

“a” is the

thing you multiply

times itself to get

the left term. In this

case a = 2x because

(2x)

2

= 4x

2

. “b” is the

thing you multiply

times itself to get

the term that’s

subtracted.

Always

check to

see if there is a

greatest common

factor before you

try to factor an

expression any other

way. In this problem,

if you don’t factor

the 2 out rst, you

don’t end up with

a difference

of perfect

squares.

After factoring the

difference of perfect squares,

it turns out one of the factors is

ALSO a difference of perfect squares.

You can’t factor x

2

+ 4, because

there’s no formula for the SUM of

perfect squares.

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