Chapter Twenty — Rational Expressions

The Humongous Book of Algebra Problems

452

Therefore, a = 4, b = 3, and c = –9.

Multiplying and Dividing Rational Expressions

Common denominators not necessary

20.20 Simplify the expression: .

To multiply two rational expressions, divide the product of the numerators by

the product of the denominators.

Reduce the fraction to lowest terms by eliminating the common factor x from

the numerator and denominator.

20.21 Simplify the expression: .

The product of rational expressions is equal to the product of the numerators

divided by the product of the denominators.

You know

it’s not possible

to reduce the nal

answer because

the numerator and

denominator didn’t have

any factors in common

in the previous step.

Chapter Twenty — Rational Expressions

The Humongous Book of Algebra Problems

453

20.22 Simplify the expression: .

Multiply the rational expressions.

Factor the numerator and denominator and simplify the expression.

Therefore, .

20.23 Simplify the expression: .

Express the quotient as a product.

Multiply the rational expressions and simplify the result.

20.24 Simplify the expression: .

Express the quotient as a product.

You might be

wondering what

happened to the

restrictions (like

x ≠ 2) from the

beginning of the

chapter. There’s no

need to include them

unless the problem tells

you to. Keep in mind,

however, that this

equation is not true

when x = –2, x = 0,

or x = 2—when

factors you

eliminated

equal 0.

This is

introduced

in Problems

2.38–2.40. To

calculate

,

take the reciprocal

of the fraction

you’re dividing by

and change the

division symbol to

multiplication:

.

Chapter Twenty — Rational Expressions

The Humongous Book of Algebra Problems

454

Factor the quadratic expression: x

2

– 2x – 3 = (x – 3)(x + 1).

Reduce the fraction to lowest terms.

20.25 Simplify the expression: .

Express the product as a quotient and factor the expressions.

Eliminate factors shared by the numerator and denominator.

Therefore, .

20.26 Simplify the expression: .

Express the product as a quotient and factor the expressions.

Either of

these answers is

ne—whether you

expand (x + 1)

2

into

x

2

+ 2x + 1 or leave

it factored

doesn’t matter.

8x

3

+ 27 is a sum of

perfect cubes and

4x

2

– 9 is a difference

of perfect squares.

Chapter Twenty — Rational Expressions

The Humongous Book of Algebra Problems

455

Eliminate factors shared by the numerator and denominator.

Therefore, .

20.27 Simplify the expression: .

Express the quotient as a product.

Factor the quadratic expressions as well as the difference of perfect squares:

x

4

– 256 = (x

2

+ 16)(x

2

– 16). Note that the factor x

2

– 16 is a difference of perfect

squares as well. Therefore, x

4

– 256 = (x

2

+ 16)(x + 4)(x – 4).

Notice that (x – 4) is a factor of x

3

– 4x

2

+ 16x – 64.

Therefore, x

3

– 4x

2

+ 16x – 64 = (x – 4)(x

2

+ 16). Write the preceding rational

expression using the factored form of the cubic.

Therefore, .

The goal of

the problem is

to simplify the

fraction, so one of

the factors in the

numerator is probably

going to match one

of the factors of

x

3

– 4x

2

+ 16x – 64. In

other words, the rst

two factors you should

try to divide in

synthetically are

(x + 4) and (x – 4).

Chapter Twenty — Rational Expressions

The Humongous Book of Algebra Problems

456

20.28 Simplify the expression: .

According to the order of operations, multiplication and division should

be performed in the same step, from left to right. Express the quotient as a

product.

Factor the quadratic expressions and simplify.

20.29 Simplify the expression: .

According to the order of operations, multiplication must be completed before

addition. Express the product as a single fraction reduced to lowest terms.

The least common denominator of the expression is x

4

(x – 5). Rewrite the

expression using the least common denominator and simplify.

Combine the numerators of the fractions.

Divide

the rst two

fractions (by

rewriting them as a

multiplication problem

and then simplifying)

and multiply the

answer by the

fraction on

the right.

3x and 15 are

both divisible by 3, so

factor it out:

3x + 15 = 3(x + 5).

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