Mastering Multiplying and Dividing Rational Expressions

Rational Expressions
Not in Simplest Form	In Simplest Form
$\frac{x y}{x ( x - 3 ) ( y + 2 )}$	$\frac{x - 1}{x + 1}$
$\frac{x ^{4} + x ^{2}}{x ^{2} + 1}$	$\frac{x ^{3} + 7}{x ^{2} - x - 6}$

Rational Expressions

Not in Simplest Form

In Simplest Form

\frac{x y}{x ( x - 3 ) ( y + 2 )}

\frac{x - 1}{x + 1}

\frac{x ^{4} + x ^{2}}{x ^{2} + 1}

\frac{x ^{3} + 7}{x ^{2} - x - 6}

Expression	Restriction	Excluded Value(s)
$\frac{x - 1}{x + 1}$	$x + 1 \neq = 0$	$x \neq = - 1$
$\frac{x ^{3} + 7}{x ^{2} - x - 6}$	$x^{2} - x - 6 \neq = 0$	$x \neq = - 2$ and $x \neq = 3$
$\frac{x y}{x ( x - 3 ) ( y + 2 )}$	$x (x - 3) (y + 2) \neq = 0$	$x \neq = 0,$ $x \neq = 3,$ and $y \neq = - 2$
$\frac{x ^{4} + x ^{2}}{x ^{2} + 1}$	There is no real number that makes $x^{2} + 1$ zero	None

Expression

Restriction

Excluded Value(s)

\frac{x - 1}{x + 1}

x + 1 \neq = 0

x \neq = - 1

\frac{x ^{3} + 7}{x ^{2} - x - 6}

x^{2} - x - 6 \neq = 0

x \neq = - 2

and

x \neq = 3

\frac{x y}{x ( x - 3 ) ( y + 2 )}

x (x - 3) (y + 2) \neq = 0

x \neq = 0,

x \neq = 3,

and

y \neq = - 2

\frac{x ^{4} + x ^{2}}{x ^{2} + 1}

There is no real number that makes

x^{2} + 1

zero

None

Equivalent Expressions
Rational Expression	Simplified Form
$\frac{x - 3}{( x + 2 ) ( x - 3 )}, x \neq = - 2, 3$	$\frac{1}{x + 2}, x \neq = - 2, 3$
$\frac{x ^{2} + 2 x + 1}{x ^{2} - 1}, x \neq = - 1, 1$	$\frac{x + 1}{x - 1}, x \neq = - 1, 1$
$\frac{x ^{3} - 2 x ^{2} + x}{x ^{2}}, x \neq = 0$	$\frac{x ^{2} - 2 x + 1}{x}, x \neq = 0$

Equivalent Expressions

Rational Expression

Simplified Form

\frac{x - 3}{( x + 2 ) ( x - 3 )}, x \neq = - 2, 3

\frac{1}{x + 2}, x \neq = - 2, 3

\frac{x ^{2} + 2 x + 1}{x ^{2} - 1}, x \neq = - 1, 1

\frac{x + 1}{x - 1}, x \neq = - 1, 1

\frac{x ^{3} - 2 x ^{2} + x}{x ^{2}}, x \neq = 0

\frac{x ^{2} - 2 x + 1}{x}, x \neq = 0

Denominator	Restrictions on the Denominator	Restrictions on the Variable
$(x + 9)^{2}$	$(x + 9)^{2} \neq = 0$	$x \neq = - 9$
$(x + 3) (x - 3)$	$x + 3 \neq = 0$ and $x - 3 \neq = 0$	$x \neq = - 3$ and $x \neq = 3$
$(x + 9) (x + 3)$	$x + 9 \neq = 0$ and $x + 3 \neq = 0$	$x \neq = - 9$ and $x \neq = - 3$

Denominator

Restrictions on the Denominator

Restrictions on the Variable

(x + 9)^{2}

(x + 9)^{2} \neq = 0

x \neq = - 9

(x + 3) (x - 3)

x + 3 \neq = 0

and

x - 3 \neq = 0

x \neq = - 3

and

x \neq = 3

(x + 9) (x + 3)

x + 9 \neq = 0

and

x + 3 \neq = 0

x \neq = - 9

and

x \neq = - 3

	Surface Area, $S$	Volume, $V$
Type I	$\frac{x ^{3} - x ^{2}}{4 x ^{4} + 1 2 x ^{3}}$	$\frac{x ^{2} - 2 x + 1}{x ^{3} + 3 x ^{2}}$
Type II	$\frac{2 x ^{2} - 9 x - 1 8}{4 x ^{2} - 2 8 x + 2 4}$	$2 x^{2} + x - 3$

Surface Area,

S

Volume,

V

Type I

\frac{x ^{3} - x ^{2}}{4 x ^{4} + 1 2 x ^{3}}

\frac{x ^{2} - 2 x + 1}{x ^{3} + 3 x ^{2}}

Type II

\frac{2 x ^{2} - 9 x - 1 8}{4 x ^{2} - 2 8 x + 2 4}

2 x^{2} + x - 3

a To find the efficiency ratios, the ratio of the surface area to the volume will be found for each packaging type. First, the efficiency ratio of Type I will be calculated.

\frac{x ^{3} - x ^{2}}{4 x ^{4} + 1 2 x ^{3}} \div \frac{x ^{2} - 2 x + 1}{x ^{3} + 3 x ^{2}}

To divide rational expressions, multiply the first expression by the reciprocal of the second expression.

\frac{x ^{3} - x ^{2}}{4 x ^{4} + 1 2 x ^{3}} \div \frac{x ^{2} - 2 x + 1}{x ^{3} + 3 x ^{2}} ⇕ \frac{x ^{3} - x ^{2}}{4 x ^{4} + 1 2 x ^{3}} \cdot \frac{x ^{3} + 3 x ^{2}}{x ^{2} - 2 x + 1}

To multiply these expressions, begin by factoring the numerators and the denominators of each expression, if possible. Start with the first rational expression.

\frac{x ^{3} - x ^{2}}{4 x ^{4} + 1 2 x ^{3}}

FactorOut

Factor out $x^{2}$

\frac{x ^{2} ( x - 1 )}{4 x ^{4} + 1 2 x ^{3}}

FactorOut

Factor out $4 x^{3}$

\frac{x ^{2} ( x - 1 )}{4 x ^{3} ( x + 3 )}

Now the second rational expression will be factored.

\frac{x ^{3} + 3 x ^{2}}{x ^{2} - 2 x + 1}

FactorOut

Factor out $x^{2}$

\frac{x ^{2} ( x + 3 )}{x ^{2} - 2 x + 1}

▼

Factor denominator

IdPropMult

Identity Property of Multiplication

\frac{x ^{2} ( x + 3 )}{x ^{2} - 2 x ( 1 ) + 1}

WritePow

Write as a power

\frac{x ^{2} ( x + 3 )}{x ^{2} - 2 x ( 1 ) + 1 ^{2}}

ExpandNegPerfectSquare

$(a - b)^{2} = a^{2} - 2 a b + b^{2}$

\frac{x ^{2} ( x + 3 )}{( x - 1 ) ^{2}}

Next the factored rational expressions will be multiplied and the common factors canceled.

\frac{x ^{2} ( x - 1 )}{4 x ^{3} ( x + 3 )} \cdot \frac{x ^{2} ( x + 3 )}{( x - 1 ) ^{2}}

MultFrac

Multiply fractions

\frac{x ^{2} ( x - 1 ) x ^{2} ( x + 3 )}{4 x ^{3} ( x + 3 ) ( x - 1 ) ^{2}}

▼

Simplify

Write power as a product

\frac{x ^{2} ( x - 1 ) x \cdot x ( x + 3 )}{4 x ^{2} \cdot x ( x + 3 ) ( x - 1 ) ( x - 1 )}

CancelCommonFac

Cancel out common factors

\frac{x ^{2} ( x - 1 ) x \cdot x ( x + 3 )}{4 x ^{2} \cdot x ( x + 3 ) ( x - 1 ) ( x - 1 )}

SimpQuot

Simplify quotient

\frac{x}{4 ( x - 1 )}

The efficiency ratio of Type I is

\frac{x}{4 ( x - 1 )} .

The efficiency ratio of the Type II packages can be found by following a similar process.

\frac{2 x ^{2} - 9 x - 1 8}{4 x ^{2} - 2 8 x + 2 4} \div (2 x^{2} + x - 3)

To divide a rational expression by polynomial, the first expression is multiplied by the reciprocal of the polynomial.

\frac{2 x ^{2} - 9 x - 1 8}{4 x ^{2} - 2 8 x + 2 4} \div (2 x^{2} + x - 3) ⇕ \frac{2 x ^{2} - 9 x - 1 8}{4 x ^{2} - 2 8 x + 2 4} \cdot \frac{1}{2 x ^{2} + x - 3}

The table shows the steps taken to perform the multiplication.

	Multiplying Rational Expressions
Product	$\frac{2 x ^{2} - 9 x - 1 8}{4 x ^{2} - 2 8 x + 2 4} \cdot \frac{1}{2 x ^{2} + x - 3}$
Factor	$\frac{( 2 x + 3 ) ( x - 6 )}{4 ( x - 1 ) ( x - 6 )} \cdot \frac{1}{( 2 x + 3 ) ( x - 1 )}$
Multiply	$\frac{( 2 x + 3 ) ( x - 6 )}{4 ( x - 1 ) ( x - 6 ) ( 2 x + 3 ) ( x - 1 )}$
Cancel Out Common Factors	$\frac{( 2 x + 3 ) ( x - 6 )}{4 ( x - 1 ) ( x - 6 ) ( 2 x + 3 ) ( x - 1 )}$
Simplify	$\frac{1}{4 ( x - 1 ) ^{2}}$

The efficiency ratio of Type II is $\frac{1}{4 ( x - 1 ) ^{2}}$

b In Part A, the efficiency ratio of each type was found. Now, these ratios will be evaluated for

x = 2 .

	Type I	Type II
Efficiency Ratio	$\frac{x}{4 ( x - 1 )}$	$\frac{1}{4 ( x - 1 ) ^{2}}$
Substitute	$\frac{2}{4 ( 2 - 1 )}$	$\frac{1}{4 ( 2 - 1 ) ^{2}}$
Evaluate	$\frac{1}{2}$	$\frac{1}{4}$

Recall that the smaller the ratio, the more efficient the packaging. Therefore, Type II is more efficient because $\frac{1}{4} < \frac{1}{2} .$

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Catch-Up and Review

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Rational Expression

Determining Values That Make a Rational Expression Undefined

Simplifying Rational Expressions

Finding the Domain of Rational Expressions

Hint

Solution

Multiplying and Dividing Rational Expressions

Multiplying Rational Expressions

Dividing Rational Expressions

Representing Areas with Rational Expressions

Hint

Solution

Determining the Type With the Best Efficiency Ratio

Hint

Solution

Complex Fractions

Surface Area to Volume Ratio of Penguins

Hint

Solution

Finding Combined Fuel Efficiency of a New Car Model

Hint

Solution

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