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The Humongous Book of Algebra Problems by W. Michael Kelley

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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 didnt 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
youre 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
doesnt 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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