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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