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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