# 3.2: Multiply and Divide Rational Expressions - Mathematics

Learning Objectives

By the end of this section, you will be able to:

• Determine the values for which a rational expression is undefined
• Simplify rational expressions
• Multiply rational expressions
• Divide rational expressions
• Multiply and divide rational functions

Before you get started, take this readiness quiz.

1. Simplify: (dfrac{90y}{15y^2}).
If you missed this problem, review [link].
2. Multiply: (dfrac{14}{15}·dfrac{6}{35}).
If you missed this problem, review [link].
3. Divide: (dfrac{12}{10}÷dfrac{8}{25}).
If you missed this problem, review [link].

We previously reviewed the properties of fractions and their operations. We introduced rational numbers, which are just fractions where the numerators and denominators are integers. In this chapter, we will work with fractions whose numerators and denominators are polynomials. We call this kind of expression a rational expression.

RATIONAL EXPRESSION

A rational expression is an expression of the form (dfrac{p}{q}), where (p) and (q) are polynomials and (q eq 0).

Here are some examples of rational expressions:

Notice that the first rational expression listed above, (−dfrac{24}{56}), is just a fraction. Since a constant is a polynomial with degree zero, the ratio of two constants is a rational expression, provided the denominator is not zero.

We will do the same operations with rational expressions that we did with fractions. We will simplify, add, subtract, multiply, divide and use them in applications.

## Determine the Values for Which a Rational Expression is Undefined

If the denominator is zero, the rational expression is undefined. The numerator of a rational expression may be 0—but not the denominator.

When we work with a numerical fraction, it is easy to avoid dividing by zero because we can see the number in the denominator. In order to avoid dividing by zero in a rational expression, we must not allow values of the variable that will make the denominator be zero.

So before we begin any operation with a rational expression, we examine it first to find the values that would make the denominator zero. That way, when we solve a rational equation for example, we will know whether the algebraic solutions we find are allowed or not.

DETERMINE THE VALUES FOR WHICH A RATIONAL EXPRESSION IS UNDEFINED.

1. Set the denominator equal to zero.
2. Solve the equation.

EXAMPLE (PageIndex{1})

Determine the value for which each rational expression is undefined:

a. (dfrac{8a^2b}{3c}) b. (dfrac{4b−3}{2b+5}) c. (dfrac{x+4}{x^2+5x+6}).

The expression will be undefined when the denominator is zero.

a.

(egin{array} {ll} &dfrac{8a^2b}{3c} egin{array} {l} ext{Set the denominator equal to zero and solve} ext{for the variable.} end{array} &3c=0 &c=0 &dfrac{8a^2b}{3c} ext{ is undefined for }c=0 end{array} )
b.

(egin{array} {ll} &dfrac{4b-3}{2b+5} egin{array} {l} ext{Set the denominator equal to zero and solve} ext{for the variable.} end{array} &2b+5=0 &2b=-5 &b=-dfrac{5}{2} & &dfrac{4b-3}{2b+5} ext{ is undefined for }b=-dfrac{5}{2} end{array} )

c.

(egin{array} {ll} &dfrac{x+4}{x^2 + 5x + 6} egin{array} {l} ext{Set the denominator equal to zero and solve } ext{for the variable.} end{array} &x^2+5x+6=0 &(x+2)(x+3)=0 &x+2=0 ext{ or }x+3=0 &x=-2 ext{ or }x=-3 & &dfrac{x+4}{x^2+5x+6} ext{ is undefined for }x=-2 ext{ or }x=-3 end{array} )

EXAMPLE (PageIndex{2})

Determine the value for which each rational expression is undefined.

a. (dfrac{3y^2}{8x}) b. (dfrac{8n−5}{3n+1}) c. (dfrac{a+10}{a^2+4a+3})

a. (x=0)
b. (n=−dfrac{1}{3})
c. (a=−1,a=−3)

EXAMPLE (PageIndex{3})

Determine the value for which each rational expression is undefined.

a.(dfrac{4p}{5q}) b. (dfrac{y−1}{3y+2}) c. (dfrac{m−5}{m^2+m−6})

a. (q=0)
b. (y=−dfrac{2}{3})
c. (m=2,m=−3)

## Simplify Rational Expressions

A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator. Similarly, a simplified rational expression has no common factors, other than 1, in its numerator and denominator.

SIMPLIFIED RATIONAL EXPRESSION

A rational expression is considered simplified if there are no common factors in its numerator and denominator.

For example,

[ egin{array} {l} dfrac{x+2}{x+3} ext{ is simplified because there are no common factors of } x+2 ext{ and }x+3. dfrac{2x}{3x} ext{ is not simplified because x is a common factor of }2x ext{ and }3x. end{array} onumber]

We use the Equivalent Fractions Property to simplify numerical fractions. We restate it here as we will also use it to simplify rational expressions.

EQUIVALENT FRACTIONS PROPERTY

If (a), (b), and (c) are numbers where (b eq 0,c eq 0,)

[ ext {then } dfrac{a}{b}=dfrac{a·c}{b·c} ext{ and } dfrac{a·c}{b·c}=dfrac{a}{b} onumber]

Notice that in the Equivalent Fractions Property, the values that would make the denominators zero are specifically disallowed. We see (b eq 0,c eq 0) clearly stated.

To simplify rational expressions, we first write the numerator and denominator in factored form. Then we remove the common factors using the Equivalent Fractions Property.

Be very careful as you remove common factors. Factors are multiplied to make a product. You can remove a factor from a product. You cannot remove a term from a sum. Removing the (x)'s from (dfrac{x+5}{x}) would be like cancelling the (2)’s in the fraction (dfrac{2+5}{2}!)

#### How to Simplify a Rational Expression

EXAMPLE (PageIndex{4})

Simplify: (dfrac{x^2+5x+6}{x^2+8x+12})  EXAMPLE (PageIndex{5})

Simplify: (dfrac{x^2−x−2}{x^2−3x+2}).

(dfrac{x+1}{x−1},x eq 2,x eq 1)

EXAMPLE (PageIndex{6})

Simplify: (dfrac{x^2−3x−10}{x^2+x−2}).

(dfrac{x−5}{x−1},x eq −2,x eq 1)

We now summarize the steps you should follow to simplify rational expressions.

SIMPLIFY A RATIONAL EXPRESSION.

1. Factor the numerator and denominator completely.
2. Simplify by dividing out common factors.

Usually, we leave the simplified rational expression in factored form. This way, it is easy to check that we have removed all the common factors.

We’ll use the methods we have learned to factor the polynomials in the numerators and denominators in the following examples.

Every time we write a rational expression, we should make a statement disallowing values that would make a denominator zero. However, to let us focus on the work at hand, we will omit writing it in the examples.

EXAMPLE (PageIndex{7})

Simplify: (dfrac{3a^2−12ab+12b^2}{6a^2−24b^2}).

(egin{array} {ll} &dfrac{3a^2−12ab+12b^2}{6a^2−24b^2} & & egin{array} {l} ext{Factor the numerator and denominator,} ext{first factoring out the GCF.} end{array} &dfrac{3(a^2−4ab+4b^2)}{6(a^2−4b^2)} & &dfrac{3(a−2b)(a−2b)}{6(a+2b)(a−2b)} & ext{Remove the common factors of }a−2b ext{ and }3. &dfrac{cancel{3}(a−2b)cancel{(a−2b)}}{cancel{3}·2(a+2b)cancel{(a−2b)}} &dfrac{a−2b}{2(a+2b)} end{array} )

EXAMPLE (PageIndex{8})

Simplify: (dfrac{2x^2−12xy+18y^2}{3x^2−27y^2}).

(dfrac{2(x−3y)}{3(x+3y)})

EXAMPLE (PageIndex{9})

Simplify: (dfrac{5x^2−30xy+25y^2}{2x^2−50y^2}).

(dfrac{5(x−y)}{2(x+5y)})

Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors. We previously introduced opposite notation: the opposite of (a) is (−a) and (−a=−1·a).

The numerical fraction, say (dfrac{7}{−7}) simplifies to (−1). We also recognize that the numerator and denominator are opposites.

The fraction (dfrac{a}{−a}), whose numerator and denominator are opposites also simplifies to (−1).

[egin{array} {ll} ext{Let’s look at the expression }b−a. &b−a ext{Rewrite.} &−a+b ext{Factor out }–1. &−1(a−b) onumberend{array} ]

This tells us that (b−a) is the opposite of (a−b).

In general, we could write the opposite of (a−b) as (b−a). So the rational expression (dfrac{a−b}{b−a}) simplifies to (−1).

OPPOSITES IN A RATIONAL EXPRESSION

The opposite of (a−b) is (b−a).

[dfrac{a−b}{b−a}=−1 quad a eq b onumber]

An expression and its opposite divide to (−1).

We will use this property to simplify rational expressions that contain opposites in their numerators and denominators. Be careful not to treat (a+b) and (b+a) as opposites. Recall that in addition, order doesn’t matter so (a+b=b+a). So if (a eq −b), then (dfrac{a+b}{b+a}=1).

EXAMPLE (PageIndex{10})

Simplify: (dfrac{x^2−4x−32}{64−x^2}) Factor the numerator and the denominator. Recognize the factors that are opposites. Simplify. EXAMPLE (PageIndex{11})

Simplify: (dfrac{x^2−4x−52}{5−x^2})

(−dfrac{x+1}{x+5})

EXAMPLE (PageIndex{12})

Simplify: (dfrac{x^2+x−2}{1−x^2}).

(−dfrac{x+2}{x+1})

## Multiply Rational Expressions

To multiply rational expressions, we do just what we did with numerical fractions. We multiply the numerators and multiply the denominators. Then, if there are any common factors, we remove them to simplify the result.

MULTIPLICATION OF RATIONAL EXPRESSIONS

If (p), (q), (r), and (s) are polynomials where (q eq 0), (s eq 0), then

[dfrac{p}{q}·dfrac{r}{s}=dfrac{pr}{qs} onumber]

To multiply rational expressions, multiply the numerators and multiply the denominators.

Remember, throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, (x eq 0), (x eq 3), and (x eq 4.)

EXAMPLE (PageIndex{13}): How to Multiply Rational Expressions

Simplify: (dfrac{2x}{x^2−7x+12}·dfrac{x^2−9}{6x^2}).   EXAMPLE (PageIndex{14})

Simplify: (dfrac{5x}{x^2+5x+6}·dfrac{x^2−4}{10x}).

(dfrac{x−2}{2(x+3)})

EXAMPLE (PageIndex{15})

Simplify: (dfrac{9x^2}{x^2+11x+30}·dfrac{x^2−36}{3x^2}).

(dfrac{3(x−6)}{x+5})

MULTIPLY RATIONAL EXPRESSIONS.

1. Factor each numerator and denominator completely.
2. Multiply the numerators and denominators.
3. Simplify by dividing out common factors.

EXAMPLE (PageIndex{16})

Multiply: (dfrac{3a^2−8a−3}{a^2−25}·dfrac{a^2+10a+25}{3a^2−14a−5}).

(egin{array} {ll} &dfrac{3a^2−8a−3}{a^2−25}·dfrac{a^2+10a+25}{3a^2−14a−5} & egin{array} {ll} ext{Factor the numerators and denominators} ext{and then multiply.} end{array} &dfrac{(3a+1)(a−3)(a+5)(a+5)}{(a−5)(a+5)(3a+1)(a−5)} & egin{array} {l} ext{Simplify by dividing out} ext{common factors.} end{array} &dfrac{cancel{(3a+1)}(a−3)cancel{(a+5)}(a+5)}{(a−5)cancel{(a+5)}cancel{(3a+1)}(a−5)} & ext{Simplify.} &dfrac{(a−3)(a+5)}{(a−5)(a−5)} & ext{Rewrite }(a−5)(a−5) ext{ using an exponent.} &dfrac{(a−3)(a+5)}{(a−5)^2} end{array})

EXAMPLE (PageIndex{17})

Simplify: (dfrac{2x^2+5x−12}{x^2−16}·dfrac{x^2−8x+16}{2x^2−13x+15}).

(dfrac{x−4}{x−5})

EXAMPLE (PageIndex{18})

Simplify: (dfrac{4b^2+7b−2}{1−b^2}·dfrac{b^2−2b+1}{4b^2+15b−4}).

(−dfrac{(b+2)(b−1)}{(1+b)(b+4)})

## Divide Rational Expressions

Just like we did for numerical fractions, to divide rational expressions, we multiply the first fraction by the reciprocal of the second.

DIVISION OF RATIONAL EXPRESSIONS

If (p), (q), (r), and (s) are polynomials where (q eq 0), (r eq 0), (s eq 0), then

[dfrac{p}{q}÷dfrac{r}{s}=dfrac{p}{q}·dfrac{s}{r} onumber]

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

Once we rewrite the division as multiplication of the first expression by the reciprocal of the second, we then factor everything and look for common factors.

EXAMPLE (PageIndex{19}): How to Divide Rational Expressions

Divide: (dfrac{p^3+q^3}{2p^2+2pq+2q^2}÷dfrac{p^2−q^2}{6}).    EXAMPLE (PageIndex{20})

Simplify: (dfrac{x^3−8}{3x^2−6x+12}÷dfrac{x^2-4}{6}).

(dfrac{2(x^2+2x+4)}{(x+2)(x^2−2x+4)})

EXAMPLE (PageIndex{21})

Simplify: (dfrac{2z^2}{z^2−1}÷dfrac{z^3−z^2+z}{z^3+1}).

(dfrac{2z}{z−1})

DIVISION OF RATIONAL EXPRESSIONS

1. Rewrite the division as the product of the first rational expression and the reciprocal of the second.
2. Factor the numerators and denominators completely.
3. Multiply the numerators and denominators together.
4. Simplify by dividing out common factors.

Recall from Use the Language of Algebra that a complex fraction is a fraction that contains a fraction in the numerator, the denominator or both. Also, remember a fraction bar means division. A complex fraction is another way of writing division of two fractions.

EXAMPLE (PageIndex{22})

Divide: (dfrac{dfrac{6x^2−7x+2}{4x−8}}{dfrac{2x^2−7x+3}{x^2−5x+6}}).

(egin{array} {ll} &dfrac{dfrac{6x^2−7x+2}{4x−8}}{dfrac{2x^2−7x+3}{x^2−5x+6}} & ext{Rewrite with a division sign.} &dfrac{6x^2−7x+2}{4x−8}÷dfrac{2x^2−7x+3}{x^2−5x+6} & egin{array} {l} ext{Rewrite as product of first times reciprocal} ext{of second.} end{array} &dfrac{6x^2−7x+2}{4x−8}·dfrac{x^2−5x+6}{2x^2−7x+3} & egin{array} {l} ext{Factor the numerators and the} ext{denominators, and then multiply.} end{array} &dfrac{(2x−1)(3x−2)(x−2)(x−3)}{4(x−2)(2x−1)(x−3)} & ext{Simplify by dividing out common factors.} &dfrac{cancel{(2x−1)}(3x−2)cancel{(x−2)}cancel{(x−3)}}{4cancel{(x−2)}cancel{(2x−1)}cancel{(x−3)}} ext{Simplify.} &dfrac{3x−2}{4} end{array})

EXAMPLE (PageIndex{23})

Simplify: (dfrac{dfrac{3x^2+7x+2}{4x+24}}{dfrac{3x^2−14x−5}{x^2+x−30}}).

(dfrac{x+2}{4})

EXAMPLE (PageIndex{24})

Simplify: (dfrac{dfrac{y^2−36}{2y^2+11y−6}}{dfrac{2y^2−2y−60}{8y−4}}).

(dfrac{2}{y+5})

If we have more than two rational expressions to work with, we still follow the same procedure. The first step will be to rewrite any division as multiplication by the reciprocal. Then, we factor and multiply.

EXAMPLE (PageIndex{26})

Perform the indicated operations: (dfrac{4m+4}{3m−15}·dfrac{m^2−3m−10}{m^2−4m−32}÷dfrac{12m−36}{6m−48}).

(dfrac{2(m+1)(m+2)}{3(m+4)(m−3)})

EXAMPLE (PageIndex{27})

Perform the indicated operations: (dfrac{2n^2+10n}{n−1}÷dfrac{n^2+10n+24}{n^2+8n−9}·dfrac{n+4}{8n^2+12n}).

(dfrac{(n+5)(n+9)}{2(n+6)(2n+3)})

## Multiply and Divide Rational Functions

We started this section stating that a rational expression is an expression of the form (dfrac{p}{q}), where p and q are polynomials and (q eq 0). Similarly, we define a rational function as a function of the form (R(x)=dfrac{p(x)}{q(x)}) where (p(x)) and (q(x)) are polynomial functions and (q(x)) is not zero.

RATIONAL FUNCTION

A rational function is a function of the form

[R(x)=dfrac{p(x)}{q(x)} onumber]

where (p(x)) and (q(x)) are polynomial functions and (q(x)) is not zero.

The domain of a rational function is all real numbers except for those values that would cause division by zero. We must eliminate any values that make (q(x)=0).

DETERMINE THE DOMAIN OF A RATIONAL FUNCTION.

1. Set the denominator equal to zero.
2. Solve the equation.
3. The domain is all real numbers excluding the values found in Step 2.

EXAMPLE (PageIndex{28})

Find the domain of (R(x)=dfrac{2x^2−14x}{4x^2−16x−48}).

The domain will be all real numbers except those values that make the denominator zero. We will set the denominator equal to zero , solve that equation, and then exclude those values from the domain.

(egin{array} {ll} ext{Set the denominator to zero.} &4x^2−16x−48=0 ext{Factor, first factor out the GCF.} &4(x^2−4x−12)=0 &4(x−6)(x+2)=0 ext{Use the Zero Product Property.} &4 eq 0quad x−6=0quad x+2=0 ext{Solve.} &hspace{24mm}x=6qquad x=−2 & ext{The domain of }R(x) ext{ is all real numbers} & ext{where }x eq 6 ext{ and }x eq −2 end{array}).

EXAMPLE (PageIndex{29})

Find the domain of (R(x)=dfrac{2x^2−10x}{4x^2−16x−20}).

The domain of (R(x)) is all real numbers where (x eq 5) and (x eq −1).

EXAMPLE (PageIndex{30})

Find the domain of (R(x)=dfrac{4x^2−16x}{8x^2−16x−64}).

The domain of (R(x)) is all real numbers where (x eq 4) and (x eq −2).

To multiply rational functions, we multiply the resulting rational expressions on the right side of the equation using the same techniques we used to multiply rational expressions.

EXAMPLE (PageIndex{31})

Find (R(x)=f(x)·g(x)) where (f(x)=dfrac{2x−6}{x^2−8x+15}) and (g(x)=dfrac{x^2−25}{2x+10}).

(egin{array} {ll} &R(x)=f(x)·g(x) & &R(x)=dfrac{2x−6}{x^2−8x+15}·dfrac{x^2−25}{2x+10} & ext{Factor each numerator and denominator.} &R(x)=dfrac{2(x−3)}{(x−3)(x−5)}·dfrac{(x−5)(x+5)}{2(x+5)} & ext{Multiply the numerators and denominators.} &R(x)=dfrac{2(x−3)(x−5)(x+5)}{2(x−3)(x−5)(x+5)} & ext{Remove common factors.} &R(x)=dfrac{cancel{2}cancel{(x−3)}cancel{(x−5)}cancel{(x+5)}}{cancel{2}cancel{(x−3)}cancel{(x−5)}cancel{(x+5)}} & ext{Simplify.} &R(x)=1 end{array})

EXAMPLE (PageIndex{32})

Find (R(x)=f(x)·g(x)) where (f(x)=dfrac{3x−21}{x^2−9x+14}) and (g(x)=dfrac{2x^2−8}{3x+6}).

(R(x)=2)

EXAMPLE (PageIndex{33})

Find (R(x)=f(x)·g(x)) where (f(x)=dfrac{x^2−x}{3x^2+27x−30}) and (g(x)=dfrac{x^2−100}{x^2−10x}).

(R(x)=dfrac{1}{3})

To divide rational functions, we divide the resulting rational expressions on the right side of the equation using the same techniques we used to divide rational expressions.

EXAMPLE (PageIndex{34})

Find (R(x)=dfrac{f(x)}{g(x)}) where (f(x)=dfrac{3x^2}{x^2−4x}) and (g(x)=dfrac{9x^2−45x}{x^2−7x+10}).

(egin{array} {ll} &R(x)=dfrac{f(x)}{g(x)} ext{Substitute in the functions }f(x),space g(x). &R(x)=dfrac{dfrac{3x^2}{x^2−4x}}{dfrac{9x^2−45x}{x^2−7x+10}} & egin{array} {l} ext{Rewrite the division as the product of} f(x) ext{ and the reciprocal of }g(x). end{array} &R(x)=dfrac{3x^2}{x^2−4x}·dfrac{x^2−7x+10}{9x^2−45x} & egin{array} {l} ext{Factor the numerators and denominators} ext{and then multiply.} end{array} &R(x)=dfrac{3·x·x·(x−5)(x−2)}{x(x−4)·3·3·x·(x−5)} & ext{Simplify by dividing out common factors.} &R(x)=dfrac{cancel{3}·cancel{x}·cancel{x}cancel{(x−5)}(x−2)}{cancel{x}(x−4)·cancel{3}·3·cancel{x}cancel{(x−5)}} & &R(x)=dfrac{x−2}{3(x−4)} end{array})

EXAMPLE (PageIndex{35})

Find (R(x)=dfrac{f(x)}{g(x)}) where (f(x)=dfrac{2x^2}{x^2−8x}) and (g(x)=dfrac{8x^2+24x}{x^2+x−6}).

(R(x)=dfrac{x−2}{4(x−8)})

EXAMPLE (PageIndex{36})

Find (R(x)=dfrac{f(x)}{g(x)}) where (f(x)=dfrac{15x^2}{3x^2+33x}) and (g(x)=dfrac{5x−5}{x^2+9x−22}).

(R(x)=dfrac{x(x−2)}{x−1})

## Key Concepts

• Determine the values for which a rational expression is undefined.
1. Set the denominator equal to zero.
2. Solve the equation.
• Equivalent Fractions Property
If (a), (b), and (c) are numbers where (b eq 0), (c eq 0), then

• How to simplify a rational expression.
1. Factor the numerator and denominator completely.
2. Simplify by dividing out common factors.
• Opposites in a Rational Expression
The opposite of (a−b) is (b−a).

An expression and its opposite divide to (−1).

• Multiplication of Rational Expressions
If (p), (q), (r), and (s) are polynomials where (q eq 0), (s eq 0), then

• How to multiply rational expressions.
1. Factor each numerator and denominator completely.
2. Multiply the numerators and denominators.
3. Simplify by dividing out common factors.
• Division of Rational Expressions
If (p), (q), (r), and (s) are polynomials where (q eq 0), (r eq 0), (s eq 0), then

• How to divide rational expressions.
1. Rewrite the division as the product of the first rational expression and the reciprocal of the second.
2. Factor the numerators and denominators completely.
3. Multiply the numerators and denominators together.
4. Simplify by dividing out common factors.
• How to determine the domain of a rational function.
1. Set the denominator equal to zero.
2. Solve the equation.
3. The domain is all real numbers excluding the values found in Step 2.

## Glossary

rational expression
A rational expression is an expression of the form (dfrac{p}{q}), where (p) and (q) are polynomials and (q eq 0).
simplified rational expression
A simplified rational expression has no common factors, other than (1), in its numerator and denominator.
rational function
A rational function is a function of the form (R(x)=dfrac{p(x)}{q(x)}) where (p(x)) and (q(x)) are polynomial functions and (q(x)) is not zero.

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## How to divide the ordinary fractions?

Example of rational numbers include:
2/3, 5/8, -3/14, -11/-5, 7/-9, 7/-15 and -6/-11 etc.

Division of ordinary fractions is done by multiplying the first fraction by the reciprocal of the second fraction. For example, to divide, 4/3 ÷ 2/3, you find the product of the first fraction and the inverse of the second fraction 4/3 x 3/2 = 2.

Other examples of dividing rational numbers are:

9/16 ÷ 5/8 = 9/16 × 8/5
= (9 × 8)/ (16 × 5) = 72/80
= 9/10
-6/25 ÷ 3/5 = -6/25 × 5/3
= <(-6) × 5>/(25 × 3)
= -30/75
= -2/5

## Multiplying and Dividing Rational Functions

The product and quotient of two rational functions can be simplified using the techniques described in this section. The restrictions to the domain of a product consist of the restrictions of each function.

Example 12: Calculate ( f ⋅ g ) ( x ) and determine the restrictions to the domain. Solution: In this case, the domain of f ( x ) consists of all real numbers except 0, and the domain of g ( x ) consists of all real numbers except 1/4. Therefore, the domain of the product consists of all real numbers except 0 and 1/4. Multiply the functions and then simplify the result. Answer: ( f ⋅ g ) ( x ) = − 4 x + 1 5 x , where x ≠ 0 , 1 4

The restrictions to the domain of a quotient will consist of the restrictions of each function as well as the restrictions on the reciprocal of the divisor.

Example 13: Calculate ( f / g ) ( x ) and determine the restrictions.  In this case, the domain of f ( x ) consists of all real numbers except 3 and 8, and the domain of g ( x ) consists all real numbers except 3. In addition, the reciprocal of g ( x ) has a restriction of −8. Therefore, the domain of this quotient consists of all real numbers except 3, 8, and −8.

Answer: ( f / g ) ( x ) = 1 , where x ≠ 3 , 8 , − 8

### Key Takeaways

• After multiplying rational expressions, factor both the numerator and denominator and then cancel common factors. Make note of the restrictions to the domain. The values that give a value of 0 in the denominator are the restrictions.
• To divide rational expressions, multiply by the reciprocal of the divisor.
• The restrictions to the domain of a product consist of the restrictions to the domain of each factor.
• The restrictions to the domain of a quotient consist of the restrictions to the domain of each rational expression as well as the restrictions on the reciprocal of the divisor.

### Topic Exercises

Part A: Multiplying Rational Expressions

Multiply. (Assume all denominators are nonzero.)

3. 5 x 2 2 y ⋅ 4 y 2 15 x 3

4. 16 a 4 7 b 2 ⋅ 49 b 32 a 3

5. x − 6 12 x 3 ⋅ 24 x 2 x − 6

6. x + 10 2 x − 1 ⋅ x − 2 x + 10

8. y 2 − 9 y + 3 ⋅ 2 y − 3 y − 3

9. 2 a − 5 a − 5 ⋅ 2 a + 5 4 a 2 − 25

10. 2 a 2 − 9 a + 4 a 2 − 16 ⋅ ( a 2 + 4 a )

11. 2 x 2 + 3 x − 2 ( 2 x − 1 ) 2 ⋅ 2 x x + 2

12. 9 x 2 + 19 x + 2 4 − x 2 ⋅ x 2 − 4 x + 4 9 x 2 − 8 x − 1

13. x 2 + 8 x + 16 16 − x 2 ⋅ x 2 − 3 x − 4 x 2 + 5 x + 4

14. x 2 − x − 2 x 2 + 8 x + 7 ⋅ x 2 + 2 x − 15 x 2 − 5 x + 6

16. 2 x − 1 x − 1 ⋅ x + 6 1 − 2 x

19. 100 − y 2 y − 10 ⋅ 25 y 2 y + 10

20. 3 y 3 6 y − 5 ⋅ 36 y 2 − 25 5 + 6 y

21. 3 a 2 + 14 a − 5 a 2 + 1 ⋅ 3 a + 1 1 − 9 a 2

22. 4 a 2 − 16 a 4 a − 1 ⋅ 1 − 16 a 2 4 a 2 − 15 a − 4

23. x + 9 − x 2 + 14 x − 45 ⋅ ( x 2 − 81 )

24. 1 2 + 5 x ⋅ ( 25 x 2 + 20 x + 4 )

25. x 2 + x − 6 3 x 2 + 15 x + 18 ⋅ 2 x 2 − 8 x 2 − 4 x + 4

26. 5 x 2 − 4 x − 1 5 x 2 − 6 x + 1 ⋅ 25 x 2 − 10 x + 1 3 − 75 x 2

Part B: Dividing Rational Expressions

Divide. (Assume all denominators are nonzero.)

29. 5 x 9 3 y 3 25 x 10 9 y 5

30. 12 x 4 y 2 21 z 5 6 x 3 y 2 7 z 3

31. ( x − 4 ) 2 30 x 4 ÷ x − 4 15 x

32. 5 y 4 10 ( 3 y − 5 ) 2 ÷ 10 y 5 2 ( 3 y − 5 ) 3

35. ( a − 8 ) 2 2 a 2 + 10 a ÷ a − 8 a

36. 24 a 2 b 3 ( a − 2 b ) ÷ 12 a b ( a − 2 b ) 5

37. x 2 + 7 x + 10 x 2 + 4 x + 4 ÷ 1 x 2 − 4

38. 2 x 2 − x − 1 2 x 2 − 3 x + 1 ÷ 1 4 x 2 − 1

39. y + 1 y 2 − 3 y ÷ y 2 − 1 y 2 − 6 y + 9

40. 9 − a 2 a 2 − 8 a + 15 ÷ 2 a 2 − 10 a a 2 − 10 a + 25

41. a 2 − 3 a − 18 2 a 2 − 11 a − 6 ÷ a 2 + a − 6 2 a 2 − a − 1

42. y 2 − 7 y + 10 y 2 + 5 y − 14 2 y 2 − 9 y − 5 y 2 + 14 y + 49

43. 6 y 2 + y − 1 4 y 2 + 4 y + 1 3 y 2 + 2 y − 1 2 y 2 − 7 y − 4

44. x 2 − 7 x − 18 x 2 + 8 x + 12 ÷ x 2 − 81 x 2 + 12 x + 36

45. 4 a 2 − b 2 b + 2 a ÷ ( b − 2 a ) 2

46. x 2 − y 2 y + x ÷ ( y − x ) 2

47. 5 y 2 ( y − 3 ) 4 x 3 ÷ 25 y ( 3 − y ) 2 x 2

48. 15 x 3 3 ( y + 7 ) ÷ 25 x 6 9 ( 7 + y ) 2

50. 3 x − 2 2 x + 1 ÷ 2 − 3 x 3 x

51. ( 7 x − 1 ) 2 4 x + 1 ÷ 28 x 2 − 11 x + 1 1 − 4 x

52. 4 x ( x + 2 ) 2 ÷ 2 − x x 2 − 4

54. ( a − 2 b ) 2 2 b ÷ ( 2 b 2 + a b − a 2 )

55. x 2 − 6 x + 9 x 2 + 7 x + 12 ÷ 9 − x 2 x 2 + 8 x + 16

56. 2 x 2 − 9 x − 5 25 − x 2 ÷ 1 − 4 x + 4 x 2 − 2 x 2 − 9 x + 5

57. 3 x 2 − 16 x + 5 100 − 4 x 2 9 x 2 − 6 x + 1 3 x 2 + 14 x − 5

58. 10 x 2 − 25 x − 15 x 2 − 6 x + 9 9 − x 2 x 2 + 6 x + 9

Recall that multiplication and division are to be performed in the order they appear from left to right. Simplify the following.

59. 1 x 2 ⋅ x − 1 x + 3 ÷ x − 1 x 3

60. x − 7 x + 9 ⋅ 1 x 3 ÷ x − 7 x

61. x + 1 x − 2 ÷ x x − 5 ⋅ x 2 x + 1

62. x + 4 2 x + 5 ÷ x − 3 2 x + 5 ⋅ x + 4 x − 3

63. 2 x − 1 x + 1 ÷ x − 4 x 2 + 1 ⋅ x − 4 2 x − 1

64. 4 x 2 − 1 3 x + 2 ÷ 2 x − 1 x + 5 ⋅ 3 x + 2 2 x + 1

Part C: Multiplying and Dividing Rational Functions

Calculate ( f ⋅ g ) ( x ) and determine the restrictions to the domain.

65. f ( x ) = 1 x and g ( x ) = 1 x − 1

66. f ( x ) = x + 1 x − 1 and g ( x ) = x 2 − 1

67. f ( x ) = 3 x + 2 x + 2 and g ( x ) = x 2 − 4 ( 3 x + 2 ) 2

68. f ( x ) = ( 1 − 3 x ) 2 x − 6 and g ( x ) = ( x − 6 ) 2 9 x 2 − 1

69. f ( x ) = 25 x 2 − 1 x 2 + 6 x + 9 and g ( x ) = x 2 − 9 5 x + 1

70. f ( x ) = x 2 − 49 2 x 2 + 13 x − 7 and g ( x ) = 4 x 2 − 4 x + 1 7 − x

Calculate ( f / g ) ( x ) and state the restrictions.

71. f ( x ) = 1 x and g ( x ) = x − 2 x − 1

72. f ( x ) = ( 5 x + 3 ) 2 x 2 and g ( x ) = 5 x + 3 6 − x

73. f ( x ) = 5 − x ( x − 8 ) 2 and g ( x ) = x 2 − 25 x − 8

74. f ( x ) = x 2 − 2 x − 15 x 2 − 3 x − 10 and g ( x ) = 2 x 2 − 5 x − 3 x 2 − 7 x + 12

75. f ( x ) = 3 x 2 + 11 x − 4 9 x 2 − 6 x + 1 and g ( x ) = x 2 − 2 x + 1 3 x 2 − 4 x + 1

76. f ( x ) = 36 − x 2 x 2 + 12 x + 36 and g ( x ) = x 2 − 12 x + 36 x 2 + 4 x − 12

Part D: Discussion Board Topics

77. In the history of fractions, who is credited for the first use of the fraction bar?

## Multiplication and Division of Radical Expressions

Note&emsp&emsp&emsp&emspThe final radical must be in standard form.

&emsp&emspTo multiply one radical by a radical expression of more than one term, we use the distributive law: a(b+c)=ab+ac .

EXAMPLE&emsp&emspMultiply 3root(2)(5root(6)-2root(10)) and simplify.

Solution&emsp&emsp 3root(2)(5root(6)-2root(10))

EXAMPLE&emsp&emspMultiply 2root(3xy)(4root(x)-3root(y)) and simplify.

Solution&emsp&emsp 2root(3xy)(4root(x)-3root(y))

&emsp&emspTo multiply two radical expressions, each with more than one term, follow the same arrangement as in multiplying polynomials.

EXAMPLE&emsp&emspMultiply (2root(3)-4root(2)) by (3root(3)+root(2)) and simplify.

&emsp&emsp&emsp&emsp&emsp&emsp EXAMPLE&emsp&emspMultiply root(3x)-root(2y) by 5root(3x)+2root(2y) and simplify. &emsp&emsp&emsp&emsp&emsp&emsp

EXAMPLE&emsp&emspExpand (root(x+3)+root(x-2))^2 and simplify. &emsp&emsp&emsp&emsp&emsp&emsp

Let&rsquos see how our math solver simplifies this and similar problems. Click on "Solve Similar" button to see more examples.

&emsp&emsp&emsp&emsp&emsp&emsp (root(a)+root(b))^2=(root(a)+root(b))(root(a)+root(b))=a+2root(ab)+b &emsp&emspWhen the radicals have different indices, we apply the rule root(n,a^m)=root(nk,a^mk) to make the indices the same as their LCM and then apply root(n,a)root(n,b)=root(n,ab) .

EXAMPLES&emsp&emsp1. root(3)root(3,3^2) = root(6,3^3)root(6,3^7)=root(6,3^7)=3root(6,3)

THEOREM&emsp&emspWhen a,b &isin R,a>0,b>0 , and n &isin N , then root(n,a)/root(n,b)=root(n,a/b) .

&emsp&emspRadical expressions can be divided according to the above theorem only when the radical indices are the same. For different radical indices, the preliminary step to make them the same must be carried out.

EXAMPLES&emsp&emsp1. root(15)/root(5)=root(15/5)=root(3)

&emsp&emsp Sometimes the numerator of a fractional radicand is not an exact multiple of the denominator, for example root(3/2) To simplify such a radical, multiply both numerator and denominator of the radicand by the smallest number that will make the denominator a perfect root.

Note&emsp&emsp&emsp&emspThe denominator is a perfect root if the exponent of each factor is an integral multiple of the radical index.

&emsp&emspTo simplify root(3/2) multiply the numerator and denominator of the radicand by 2 .

&emsp&emspIt is easier to manipulate 1/2 root(6) than root(3/2) .

Remark&emsp&emspWhen the radical expression is of the form a/(b&radicc) , multiply the numerator and the denominator by root(c) .

EXAMPLE&emsp&emspDivide root(15) by root(21) and put in standard form.

Solution&emsp&emsp root(15)/root(21)=root(15/21)=root(5/7)=root((5*7)/(7*7))=1/7root(35)

EXAMPLE&emsp&emspDivide root(3xy) by root(4a^3b) and put in standard form.

Solution&emsp&emsp root(3xy)/root(4a^3b)=root((3xy)/(2^2a^3b))=root((3xy)/(2^2a^4b)*(ab)/(ab))

EXAMPLE&emsp&emspPut root((3a^2b^3)/(20xy^5)) in standard form.

Solution&emsp&emsp root((3a^2b^3)/(20xy^5)) = root((3a^2b^3)/(2^2*5xy^5)*(5xy)/(5xy)

EXAMPLE&emsp&emspDivide root(3,3) by root(3,20) and put in standard form.

Solution&emsp&emsp root(3,3)/root(3,20)=root(3,3/(2^2*5))=root(3,3/(2^2*5)*(2*5^2)/(2*5^2)

EXAMPLE&emsp&emspPut root(3,(81x^6y^7)/(8a^8b^10) in standard form.

Solution&emsp&emsp root(3,(81x^6y^7)/(8a^8b^10) = root(3,(3^4x^6y^7)/(2^3a^8b^10))=root(3,(3^4x^6y^7)/(2^3a^8b^10)*(ab^2)/(ab^2))

&emsp&emspThe deﬁnition of addition of fractions, (a+b)/c=a/c+b/c , is used to divide a radical expression with more than one term by a one-term radical.

EXAMPLE&emsp&emspDivide and simplify (3root(6)-6root(10))/(3root(2)) .

Solution&emsp&emsp (3root(6)-6root(10))/(3root(2))

EXAMPLE&emsp&emspDivide and simplify (root(7x)-root(2y))/root(14xy) .

Solution&emsp&emsp (root(7x)-root(2y))/root(14xy)

&emsp&emspWhen we multiply the radical expressions (root(a)+root(b)) and (root(a)-root(b)) , we have get the rational expression (a-b) . Each of the expressions (root(a)+root(b)) and (root(a)-root(b)) is called a rationalizing factor of other.

EXAMPLES&emsp&emsp1. root(2)-root(3) is a rationalizing factor of root(2)+root(3) .

&emsp&emsp&emsp&emsp&emsp&emsp 2. 2+3root(2) is a rationalizing factor of 2-3root(2) .

&emsp&emspTo facilitate the manipulation with a radical expression such as (root(2)+root(3))/(2root(2)+root(3)) , we change the fraction to an equivalent one with a rational denominator. This can be accomplished by multiplying both numerator and denominator by the rationalizing factor of the denominator, 2root(2)-root(3) .

&emsp&emspThis operation is called rationalizing the denominator

EXAMPLE&emsp&emspRationalize the denominator of root(2)/(2-root(3)) .

Solution&emsp&emsp root(2)/(2-root(3))

EXAMPLE&emsp&emspRationalize the denominator of (root(2)+root(3))/(2root(2)+root(3)) .

Solution&emsp&emsp (root(2)+root(3))/(2root(2)+root(3))

Let&rsquos see how our math solver simplifies this and similar problems. Click on "Solve Similar" button to see more examples.

## Multiplying and Dividing Rational Expressions - Problem 3 Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Simplifying a rational expression with a lot going on. So what we have here are three fractions all with polynomials that are either being multiplied here or divided over there. And what we really want to do is develop a sort of methodical approach into how we simplify this. So what I’m going to do is just go down the row and first take out any common factors we have from any of these terms.

So most things we could be able to factorize here are by making the numbers smaller or the powers in x smaller it’s going to make our life significantly easier. So for this first expression what I see is I can factor out a x leaving us with 2x² plus 3x minus 2. For our second one we can factor out our x² leaving us with 2x minus 1, continuing down the row, nothing I can do with this guy we can take out a 3 from this one leaving us with an x minus 5. From this one we could take out a 3 from the numerator leaving us with x² plus 4x plus 4 and lastly we can take out a 5x leaving us with x minus 2.

So we just took out common factors to begin with and now what I’m going to do is combine a couple steps. So what I want to do is I want to rewrite this. First as I go through factoring each of these out and then also turning my division sign into a multiplication by flipping this last expression over.

So going through this what we end up with is x and then we want to end up with a positive 3x so that will be a 2x plus 2 and then a minus 1. You can always double check we end up with 4x minus x just 3x and the -2 in our single term. X², 2x minus 1, so we factored the first expression. Times we want to factor to this expression as well, so this is x plus 2, x minus 5, noting we can do with this statement nothing we can do with this statement. 3 x minus 5 and our last statement turning into multiplication this means we are going to flip this over so our 5x times x minus 2 is going to come into the numerator and our 3 times, this turns into what plus 2², ends up with a denominator.

Now I’ve drawn my parenthesis which is perfectly okay because we are actually multiplying three things together and remember if we are multiplying order doesn’t matter so we could either multiply these two out first and then by a third or we can just do these two out. The order doesn’t matter for multiplying do I will just drop my parenthesis all together.

So now what I’m looking for is just common factors that we can cancel out. So let’s see what I have, we have 2x minus 1, 2x minus 1 that can cancel. X plus 2 cancels with one of these x plus 2’s, this x plus 2 cancels with the other one, x minus 5, x minus 5. Anything else that can cancel we have an x² down here and an x and an x so those can all cancel and what I have left in the top then is just a 5x minus 2 and just a 3 and a 3 in the bottom. So leaving us with 5 (x minus 2) strike that a little bit clear for you, and then just a 3 and a 3 in the bottom so that’s over 9.

So a lot of work in order to do there’s a lot of factoring going on but basically all we did was simplified each thing and each term up individually, factored it out cancelled by factors and then we are just left with our simplified form.

As with most work with rational expressions, knowledge and comfort with fractions are important to ensuring success on this exercise.

1. The general process to combine rational expressions here is to factor the numerators and denominators, multiply or divide as requested, and then reduce common factors from top and bottom as possible.
2. The denominator should be left in a factored form with GCFs factored out, especially from the denominator.
3. Like terms and factors do need to be reduced from the top and bottom.

## Questions

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. ## 3.2: Multiply and Divide Rational Expressions - Mathematics

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