2.1 Multiply and Divide Rational Expressions
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
Try It
Before you get started, take this readiness quiz.
1) Simplify: [latex]\frac{90y}{15{y}^{2}}[/latex]
2) Multiply: [latex]\frac{14}{15}·\frac{6}{35}[/latex]
3) Divide: [latex]\frac{12}{10}÷\frac{8}{25}[/latex]
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 [latex]\frac{p}{q},[/latex] where [latex]p[/latex] and [latex]q[/latex] are polynomials and [latex]q\ne 0[/latex].
Here are some examples of rational expressions:
Notice that the first rational expression listed above, [latex]-\frac{24}{56}[/latex], 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 [latex]0[/latex]—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.
HOW TO
Determine the values for which a rational expression is undefined.
- Set the denominator equal to zero.
- Solve the equation.
Examples 2.1.1
Determine the value for which each rational expression is undefined:
a. [latex]\frac{8{a}^{2}b}{3c}[/latex]
b. [latex]\frac{4b-3}{2b+5}[/latex]
c. [latex]\frac{x+4}{{x}^{2}+5x+6}[/latex]
The expression will be undefined when the denominator is zero.
Solution
a.
Step 1: Set the denominator equal to zero and solve for the variable.
[latex]\begin{align*}3c&=0\\c&=0\end{align*}[/latex]
[latex]\frac{8{a}^{2}b}{3c}[/latex] is undefined for [latex]c=0[/latex].
b.
Step 1: Set the denominator equal to zero and solve for the variable.
[latex]\begin{align*}2b+5&=0\\2b&=-5\\b&=-\frac52\end{align*}[/latex]
[latex]\frac{4b-3}{2b+5}[/latex] is undefined for [latex]b=-\frac52[/latex]
c.
Step 1: Set the denominator equal to zero and solve for the variable.
[latex]\begin{align*}x^2+5x+6&=0\\(x+2)(x+3)&=0\\x+2=0\;\text{or}\;x+3&=0\\x=-2\;\text{or}\;x&=-3\end{align*}[/latex]
[latex]\frac{x+4}{{x}^{2}+5x+6}[/latex] is undefined for [latex]x=-2[/latex] or [latex]x=-3[/latex]
Try It
4) Determine the value for which each rational expression is undefined.
a. [latex]\frac{3{y}^{2}}{8x}[/latex]
b. [latex]\frac{8n-5}{3n+1}[/latex]
c. [latex]\frac{a+10}{{a}^{2}+4a+3}[/latex]
Solution
a. [latex]x=0[/latex]
c. [latex]a=-1,a=-3[/latex]
Try It
5) Determine the value for which each rational expression is undefined.
a. [latex]\frac{4p}{5q}[/latex]
b. [latex]\frac{y-1}{3y+2}[/latex]
c. [latex]\frac{m-5}{{m}^{2}+m-6}[/latex]
Solution
a. [latex]q=0[/latex]
b. [latex]y=-\frac{2}{3}[/latex]
c. [latex]m=2,m=-3[/latex]
Simplify Rational Expressions
A fraction is considered simplified if there are no common factors, other than [latex]1[/latex], in its numerator and denominator. Similarly, a simplified rational expression has no common factors, other than [latex]1[/latex], 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,
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 [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] are numbers where [latex]b\ne 0,c\ne 0,[/latex]
Notice that in the Equivalent Fractions Property, the values that would make the denominators zero are specifically disallowed. We see [latex]b\ne 0,c\ne 0[/latex] 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.
| [latex]\frac{2\cdot\cancel3\cdot\cancel7}{\cancel3\cdot5\cdot\cancel7}[/latex] | [latex]\frac{3x\cancel{(x-9)}}{5\cancel{(x-9)}}\;where\;x\neq9[/latex] | [latex]\frac{x+5}x[/latex] |
| [latex]\frac25[/latex] | [latex]\frac{3x}5[/latex] | NO COMMON FACTORS |
| We removed the common factors [latex]3[/latex] and [latex]7[/latex]. They are factors of the product. | We removed the common factor [latex](x-9)[/latex]. It is a factor of the product. | While there is an [latex]x[/latex] in both the numerator and denominator, the [latex]{\color{red}{x}}[/latex] in the numerator is a term of a sum! |
Removing the [latex]x[/latex]’s from [latex]\frac{x+5}{x}[/latex] would be like cancelling the [latex]2[/latex]’s in the fraction [latex]\frac{2+5}{2}[/latex]!
Examples 2.1.2
Simplify: [latex]\frac{{x}^{2}+5x+6}{{x}^{2}+8x+12}[/latex]
Solution
Step 1: Factor the numerator and denominator completely.
Factor [latex]x^2+5x+6[/latex] and [latex]x^2+8x+12[/latex]
[latex]\begin{align*}\frac{x^2+5x+6}{x^2+8x+12}\\[2ex]\frac{(x+2)(x+3)}{(x+2)(x+6)}\end{align*}[/latex]
Step 2: Simplify by dividing out common factors.
Remove the common factor [latex]x+2[/latex] from the numerator and the denominator.
[latex]\begin{align*}\frac{\cancel{(x+2)}(x+3)}{\cancel{(x+2)}(x+6)}\\[2ex]\frac{(x+3)}{(x+6)}\\[2ex]x\neq-2\;x\neq-6\end{align*}[/latex]
Try It
6) Simplify: [latex]\frac{{x}^{2}-x-2}{{x}^{2}-3x+2}[/latex]
Solution
[latex]\frac{x+1}{x-1}[/latex], [latex]x\ne 2,[/latex][latex]x\ne 1[/latex]
Try It
7) Simplify: [latex]\frac{{x}^{2}-3x-10}{{x}^{2}+x-2}[/latex]
Solution
[latex]\frac{x-5}{x-1}[/latex], [latex]x\ne \text{−}2,\;x\ne 1[/latex]
We now summarize the steps you should follow to simplify rational expressions.
HOW TO
Simplify a rational expression.
- Factor the numerator and denominator completely.
- 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.
Examples 2.1.3
Simplify: [latex]\frac{3{a}^{2}-12ab+12{b}^{2}}{6{a}^{2}-24{b}^{2}}[/latex]
Solution
Step 1: Factor the numerator and denominator, first factoring out the GCF.
[latex]\begin{array}{cccc}\frac{3\left({a}^{2}-4ab+4{b}^{2}\right)}{6\left({a}^{2}-4{b}^{2}\right)}\\[2ex]\frac{3\left(a-2b\right)\left(a-2b\right)}{6\left(a+2b\right)\left(a-2b\right)}\end{array}[/latex]
Step 2: Remove the common factors of [latex]a-2ab[/latex] and [latex]3[/latex].
[latex]\begin{array}{cccc} \frac{\cancel{3}\left(a-2b\right)\cancel{\left(a-2b\right)}}{\cancel{3}\cdot2\left(a+2b\right)\cancel{\left(a-2b\right)}} \\[2ex] \frac{a-2b}{2\left(a+2b\right)}\end{array}[/latex]
Try It
8) Simplify: [latex]\frac{2{x}^{2}-12xy+18{y}^{2}}{3{x}^{2}-27{y}^{2}}[/latex]
Solution
[latex]\frac{2\left(x-3y\right)}{3\left(x+3y\right)}[/latex]
Try It
9) Simplify: [latex]\frac{5{x}^{2}-30xy+25{y}^{2}}{2{x}^{2}-50{y}^{2}}[/latex]
Solution
[latex]\frac{5\left(x-y\right)}{2\left(x+5y\right)}[/latex]
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 [latex]a[/latex] is [latex]\text{−}a[/latex] and [latex]\text{−}a=-1\cdot a[/latex].
The numerical fraction, say [latex]\frac{7}{-7}[/latex] simplifies to [latex]-1[/latex]. We also recognize that the numerator and denominator are opposites.
The fraction [latex]\frac{a}{\text{−}a}[/latex], whose numerator and denominator are opposites also simplifies to [latex]-1[/latex].
[latex]\begin{array}{cccc}\text{Let’s look at the expression}\phantom{\rule{0.2em}{0ex}}b-a.\hfill & & & \hfill \phantom{\rule{2em}{0ex}}b-a\hfill \\ \text{Rewrite.}\hfill & & & \hfill \phantom{\rule{2em}{0ex}}\text{−}a+b\hfill \\ \text{Factor out}\phantom{\rule{0.2em}{0ex}}–1.\hfill & & & \hfill \phantom{\rule{2em}{0ex}}-1\left(a-b\right)\hfill \end{array}[/latex]
This tells us that [latex]b-a[/latex] is the opposite of [latex]a-b[/latex].
In general, we could write the opposite of [latex]a-b[/latex] as [latex]b-a[/latex]. So the rational expression [latex]\frac{a-b}{b-a}[/latex] simplifies to [latex]-1[/latex].
Opposites in a Rational Expression
The opposite of [latex]a-b[/latex] is [latex]b-a[/latex].
An expression and its opposite divide to [latex]-1[/latex].
We will use this property to simplify rational expressions that contain opposites in their numerators and denominators. Be careful not to treat [latex]a+b[/latex] and [latex]b+a[/latex] as opposites. Recall that in addition, order doesn’t matter so [latex]a+b=b+a[/latex]. So if [latex]a\ne \text{−}b[/latex], then [latex]\frac{a+b}{b+a}=1[/latex].
Examples 2.1.4
Simplify: [latex]\frac{{x}^{2}-4x-32}{64-{x}^{2}}[/latex]
Solution
Step 1: Factor the numerator and the denominator.
[latex]\frac{(x-8)(x+4)}{(8-x)(8+x)}[/latex]
Step 2: Recognize the factors that are opposites.
Try It
10) Simplify: [latex]\frac{{x}^{2}-4x-5}{25-{x}^{2}}[/latex]
Solution
[latex]-\frac{x+1}{x+5}[/latex]
Try It
11) Simplify: [latex]\frac{{x}^{2}+x-2}{1-{x}^{2}}[/latex]
Solution
[latex]-\frac{x+2}{x+1}[/latex]
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 [latex]p[/latex], [latex]q[/latex], [latex]r[/latex], and [latex]s[/latex] are polynomials where [latex]q\ne 0,s\ne 0,[/latex] then
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, [latex]x\ne 0,\;x\ne 3[/latex], and [latex]x\ne 4[/latex].
Examples 2.1.5
Simplify: [latex]\frac{2x}{{x}^{2}-7x+12}·\frac{{x}^{2}-9}{6{x}^{2}}[/latex]
Solution
Step 1: Factor each numerator and denominator completely.
Factor [latex]x^2-9[/latex] and [latex]x^2-7x+12[/latex]
[latex]\begin{array}{cc}\frac{2x}{x^2-7x+12}\cdot \frac{x^2-9}{6x^2}\\[2ex] \frac{2x}{(x-3)(x-4)}\cdot \frac{(x-3)(x+3)}{6x^2}\end{array}[/latex]
Step 2: Multiply the numerators and denominators.
Multiply the numerators and denominators. It is helpful to write the monomials first.
[latex]\frac{2x(x-3)(x+3)}{6x^2(x-3)(x-4)}[/latex]
Step 3: Simplify by dividing out common factors.
Divide out the common factors.
Leave the denominator in factored form.
[latex]\begin{array}{cc}\frac{\cancel{2}\cancel{x}(\cancel{x-3})(x+3)}{\cancel{2}\cdot3\cdot \cancel{x}\cdot x(\cancel{x-3})(x-4)}\\[2ex]\frac{(x+3)}{3x(x-4)}\end{array}[/latex]
Try It
12) Simplify: [latex]\frac{5x}{{x}^{2}+5x+6}·\frac{{x}^{2}-4}{10x}[/latex]
Solution
[latex]\frac{x-2}{2\left(x+3\right)}[/latex]
Try It
13) Simplify: [latex]\frac{9{x}^{2}}{{x}^{2}+11x+30}·\frac{{x}^{2}-36}{3{x}^{2}}[/latex]
Solution
[latex]\frac{3\left(x-6\right)}{x+5}[/latex]
HOW TO
Multiply rational expressions
- Factor each numerator and denominator completely.
- Multiply the numerators and denominators.
- Simplify by dividing out common factors.
Examples 2.1.6
Multiply: [latex]\frac{3{a}^{2}-8a-3}{{a}^{2}-25}·\frac{{a}^{2}+10a+25}{3{a}^{2}-14a-5}[/latex]
Solution
Step 1: Factor the numerators and denominators and then multiply.
[latex]\frac{\left(3a+1\right)\left(a-3\right)\left(a+5\right)\left(a+5\right)}{\left(a-5\right)\left(a+5\right)\left(3a+1\right)\left(a-5\right)}[/latex]
Step 2: Simplify by dividing out common factors.
[latex]\frac{\cancel{\left(3a+1\right)}\left(a-3\right)\cancel{\left(a+5\right)}\left(a+5\right)}{\left(a-5\right)\cancel{\left(a+5\right)}\cancel{\left(3a+1\right)}\left(a-5\right)}[/latex]
Step 3: Simplify.
[latex]\frac{\left(a-3\right)\left(a+5\right)}{\left(a-5\right)\left(a-5\right)}[/latex]
Step 4: Rewrite [latex](a-5)(a-5)[/latex] using an exponent.
[latex]\frac{\left(a-3\right)\left(a+5\right)}{{\left(a-5\right)}^{2}}[/latex]
Try It
14) Simplify: [latex]\frac{2{x}^{2}+5x-12}{{x}^{2}-16}·\frac{{x}^{2}-8x+16}{2{x}^{2}-13x+15}[/latex]
Solution
[latex]\frac{x-4}{x-5}[/latex]
Try It
15) Simplify: [latex]\frac{4{b}^{2}+7b-2}{1-{b}^{2}}·\frac{{b}^{2}-2b+1}{4{b}^{2}+15b-4}[/latex]
Solution
[latex]-\frac{\left(b+2\right)\left(b-1\right)}{\left(1+b\right)\left(b+4\right)}[/latex]
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 [latex]p[/latex], [latex]q[/latex], [latex]r[/latex], and [latex]s[/latex] are polynomials where [latex]q\ne 0,r\ne 0,s\ne 0[/latex], then
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.
Examples 2.1.7
Divide: [latex]\frac{{p}^{3}+{q}^{3}}{2{p}^{2}+2pq+2{q}^{2}}÷\frac{{p}^{2}-{q}^{2}}{6}[/latex]
Solution
Step 1: Rewrite the division as the product of the first rational expression and the reciprocal of the second.
“Flip” the second fraction and change the division sign to multiplication.
[latex]\begin{align*}\frac{p^3+q^3}{2p^2+2pq+2q^2}\;&\div\;\frac{p^2-q^2}{6}\\[2ex]\frac{p^3+q^3}{2p^2+2pq+2q^2}\;&\times\;\frac{6}{p^2-q^2}\end{align*}[/latex]
Step 2: Factor the numerators and denominators completely.
Factor the numerators and denominators.
[latex]\frac{(p+q)(p^2-pq+q^2)}{2(p^2+pq+q^2)}\times\frac{2\times3}{(p-q)(p+q)}[/latex]
Step 3: Multiply the numerators and denominators.
Multiply the numerators and multiply the denominators.
[latex]\frac{(p+q)(p^2-pq+q^2)2\times3}{2(p^2+pq+q^2)(p-q)(p+q)}[/latex]
Step 4: Simplify by dividing out common factors.
Divide out the common factors.
[latex]\begin{array}{cc}\frac{\cancel{(p+q)}(p^2-pq+q^2)\cancel{2}\times3}{\cancel{2}(p^2+pq+q^2)(p-q)\cancel{(p+q)}}\\[2ex]\frac{3(p^2-pq+q^2)}{(p-q)(p^2+pq+q^2)}\end{array}[/latex]
Try It
16) Simplify: [latex]\frac{{x}^{3}-8}{3{x}^{2}-6x+12}÷\frac{{x}^{2}-4}{6}[/latex]
Solution
[latex]\frac{2\left({x}^{2}+2x+4\right)}{\left(x+2\right)\left({x}^{2}-2x+4\right)}[/latex]
Try It
17) Simplify: [latex]\frac{2{z}^{2}}{{z}^{2}-1}÷\frac{{z}^{3}-{z}^{2}+z}{{z}^{3}+1}[/latex]
Solution
[latex]\frac{2z}{z-1}[/latex]
HOW TO
Divide Rational Expressions.
- Rewrite the division as the product of the first rational expression and the reciprocal of the second.
- Factor the numerators and denominators completely.
- Multiply the numerators and denominators together.
- 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.
Examples 2.1.8
Divide: [latex]\frac{\frac{6{x}^{2}-7x+2}{4x-8}}{\frac{2{x}^{2}-7x+3}{{x}^{2}-5x+6}}[/latex]
Solution
Step 1: Rewrite with a division sign.
[latex]\frac{6{x}^{2}-7x+2}{4x-8}\div\frac{2{x}^{2}-7x+3}{{x}^{2}-5x+6}[/latex]
Step 2: Rewrite as product of first times reciprocal of second.
[latex]\frac{6{x}^{2}-7x+2}{4x-8}\times\frac{{x}^{2}-5x+6}{2{x}^{2}-7x+3}[/latex]
Step 3: Factor the numerators and the denominators, and then multiply.
[latex]\frac{\left(2x-1\right)\left(3x-2\right)\left(x-2\right)\left(x-3\right)}{4\left(x-2\right)\left(2x-1\right)\left(x-3\right)}[/latex]
Step 4: Simplify by dividing out common factors.
[latex]\begin{array}{ccc}&\;&\frac{\cancel{\left(2x-1\right)}\left(3x-2\right)\cancel{\left(x-2\right)}\cancel{\left(x-3\right)}}{4\cancel{\left(x-2\right)}\cancel{\left(2x-1\right)}\cancel{\left(x-3\right)}}\\[3ex] &\text{Simplify.}\;\;\;&\frac{3x-2}{4}\end{array}[/latex]
Try It
18) Simplify: [latex]\frac{\frac{3{x}^{2}+7x+2}{4x+24}}{\frac{3{x}^{2}-14x-5}{{x}^{2}+x-30}}[/latex]
Solution
[latex]\frac{x+2}{4}[/latex]
Try It
19) Simplify: [latex]\frac{\frac{{y}^{2}-36}{2{y}^{2}+11y-6}}{\frac{2{y}^{2}-2y-60}{8y-4}}[/latex]
Solution
[latex]\frac{2}{y+5}[/latex]
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.
Examples 2.1.9
Perform the indicated operations: [latex]\frac{3x-6}{4x-4}·\frac{{x}^{2}+2x-3}{{x}^{2}-3x-10}÷\frac{2x+12}{8x+16}[/latex]
Solution
Step 1: Rewrite the division as multiplication by the reciprocal.
[latex]\frac{3x-6}{4x-4}\times\frac{{x}^{2}+2x-3}{{x}^{2}-3x-10}\times\frac{{\color{red}{8x+16}}}{{\color{red}{2x+12}}}[/latex]
Step 2: Factor the numerators and the denominators.
[latex]\frac{3(x-2)}{4(x-1)}\times\frac{(x+3)(x-1)}{(x+2)(x-5)}\times\frac{8(x+2)}{2(x+6)}[/latex]
Step 3: Multiply the fractions.
Bringing the constants to the front will help when removing common factors.
Simplify by dividing out common factors.
[latex]\begin{array}{ccc}&\;&\frac{3\times\cancel8(x-2)(x+3)(\cancel{x-1})(\cancel{x+2})}{\cancel4\times\cancel{\cancel2(x-1)}(\cancel{x+2)}(x-5)(x+6)}\\[3ex]&\text{Simplify.}\;\;\;&\frac{3(x-2)(x+3)}{(x-5)(x+6)}\end{array}[/latex]
Try It
20) Perform the indicated operations: [latex]\frac{4m+4}{3m-15}\times\frac{{m}^{2}-3m-10}{{m}^{2}-4m-32}\div\frac{12m-36}{6m-48}[/latex]
Solution
[latex]\frac{2\left(m+1\right)\left(m+2\right)}{3\left(m+4\right)\left(m-3\right)}[/latex]
Try It
21) Perform the indicated operations: [latex]\frac{2{n}^{2}+10n}{n-1}\div\frac{{n}^{2}+10n+24}{{n}^{2}+8n-9}\times\frac{n+4}{8{n}^{2}+12n}[/latex]
Solution
[latex]\frac{\left(n+5\right)\left(n+9\right)}{2\left(n+6\right)\left(2n+3\right)}[/latex]
Multiply and Divide Rational Functions
We started this section stating that a rational expression is an expression of the form [latex]\frac{p}{q}[/latex], where [latex]p[/latex] and [latex]q[/latex] are polynomials and [latex]q\ne 0[/latex]. Similarly, we define a rational function as a function of the form [latex]R\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}[/latex] where [latex]p\left(x\right)[/latex] and [latex]q\left(x\right)[/latex] are polynomial functions and [latex]q\left(x\right)[/latex] is not zero.
Rational Function
A rational function is a function of the form
where [latex]p\left(x\right)[/latex] and [latex]q\left(x\right)[/latex] are polynomial functions and [latex]q\left(x\right)[/latex] 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 [latex]q\left(x\right)=0[/latex].
HOW TO
Determine the domain of a rational function.
- Set the denominator equal to zero.
- Solve the equation.
- The domain is all real numbers excluding the values found in Step 2:
Examples 2.1.10
Find the domain of [latex]R\left(x\right)=\frac{2{x}^{2}-14x}{4{x}^{2}-16x-48}[/latex]
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.
Solution
Step 1: Set the denominator to zero.
[latex]4{x}^{2}-16x-48=0[/latex]
Step 2: Factor, first factor out the GCF.
[latex]\begin{align*}4\left({x}^{2}-4x-12\right)&=0\\4\left(x-6\right)\left(x+2\right)&=0\end{align*}[/latex]
Step 3: Use the Zero Product Property.
[latex]4\ne 0\;\;\;x-6=0\;\;\;x+2=0[/latex]
Step 4: Solve.
[latex]x=6\;\;x=-2[/latex]
The domain of [latex]R(x)[/latex] is all real numbers where [latex]x\neq6[/latex] and [latex]x\neq-2[/latex].
Try It
22) Find the domain of [latex]R\left(x\right)=\frac{2{x}^{2}-10x}{4{x}^{2}-16x-20}[/latex]
Solution
The domain of [latex]R\left(x\right)[/latex] is all real numbers where [latex]x\ne 5[/latex] and [latex]x\ne -1[/latex].
Try It
23) Find the domain of [latex]R\left(x\right)=\frac{4{x}^{2}-16x}{8{x}^{2}-16x-64}[/latex]
Solution
The domain of [latex]R\left(x\right)[/latex] is all real numbers where [latex]x\ne 4[/latex] and [latex]x\ne -2[/latex].
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.
Examples 2.1.11
Find [latex]R\left(x\right)=f\left(x\right)\times g\left(x\right)[/latex] where [latex]f\left(x\right)=\frac{2x-6}{{x}^{2}-8x+15}[/latex] and [latex]g\left(x\right)=\frac{{x}^{2}-25}{2x+10}[/latex].
Solution
[latex]\begin{align*}R\left(x\right)&=f\left(x\right)\times g\left(x\right)\\R\left(x\right)&=\frac{2x-6}{{x}^{2}-8x+15}\times\frac{{x}^{2}-25}{2x+10}\end{align*}[/latex]
Step 1: Factor each numerator and denominator.
[latex]R\left(x\right)=\frac{2\left(x-3\right)}{\left(x-3\right)\left(x-5\right)}\times\frac{\left(x-5\right)\left(x+5\right)}{2\left(x+5\right)}[/latex]
Step 2: Multiply the numerators and denominators.
[latex]R\left(x\right)=\frac{2\left(x-3\right)\left(x-5\right)\left(x+5\right)}{2\left(x-3\right)\left(x-5\right)\left(x+5\right)}[/latex]
Step 3: Remove common factors.
[latex]\begin{align*} &\;\;&R\left(x\right)&=\frac{\cancel{2}\cancel{\left(x-3\right)}\cancel{\left(x-5\right)}\cancel{\left(x+5\right)}}{\cancel{2}\cancel{\left(x-3\right)}\cancel{\left(x-5\right)}\cancel{\left(x+5\right)}}\\[2ex] &\text{Simplify.}\;\;&R\left(x\right)&=1 \end{align*}[/latex]
Try It
24) Find [latex]R\left(x\right)=f\left(x\right)\times g\left(x\right)[/latex] where [latex]f\left(x\right)=\frac{3x-21}{{x}^{2}-9x+14}[/latex] and [latex]g\left(x\right)=\frac{2{x}^{2}-8}{3x+6}[/latex]
Solution
[latex]R\left(x\right)=2[/latex]
Try It
25) Find [latex]R\left(x\right)=f\left(x\right)\times g\left(x\right)[/latex] where [latex]f\left(x\right)=\frac{{x}^{2}-x}{3{x}^{2}+27x-30}[/latex] and [latex]g\left(x\right)=\frac{{x}^{2}-100}{{x}^{2}-10x}[/latex]
Solution
[latex]R\left(x\right)=\frac{1}{3}[/latex]
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.
Examples 2.1.12
Find [latex]R\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}[/latex] where [latex]f\left(x\right)=\frac{3{x}^{2}}{{x}^{2}-4x}[/latex] and [latex]g\left(x\right)=\frac{9{x}^{2}-45x}{{x}^{2}-7x+10}[/latex]
Solution
Step 1: Substitute in the functions [latex]f(x)[/latex], [latex]g(x)[/latex].
[latex]\begin{align*}R\left(x\right)&=\frac{f\left(x\right)}{g\left(x\right)}\\[2ex]R\left(x\right)&=\frac{\frac{3{x}^{2}}{{x}^{2}-4x}}{\frac{9{x}^{2}-45x}{{x}^{2}-7x+10}}\end{align*}[/latex]
Step 2: Rewrite the division as the product of [latex]f(x)[/latex] and the reciprocal of [latex]g(x)[/latex].
[latex]R\left(x\right)=\frac{3{x}^{2}}{{x}^{2}-4x}\times\frac{{x}^{2}-7x+10}{9{x}^{2}-45x}[/latex]
Step 3: Factor the numerators and denominators and then multiply.
[latex]R\left(x\right)=\frac{3\cdot x\cdot x\cdot\left(x-5\right)\left(x-2\right)}{x\left(x-4\right)\cdot3\cdot3\cdot x\cdot\left(x-5\right)}[/latex]
Step 4: Simplify by dividing out common factors.
[latex]\begin{align*}R\left(x\right)&=\frac{\cancel{3}\cdot\cancel{x}\cdot\cancel{x}\cancel{\left(x-5\right)}\left(x-2\right)}{\cancel{x}\left(x-4\right)\cdot\cancel{3}\cdot3\cdot\cancel{x}\cancel{\left(x-5\right)}}\\[2ex]R\left(x\right)&=\frac{x-2}{3\left(x-4\right)}\end{align*}[/latex]
Try It
26) Find [latex]R\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}[/latex] where [latex]f\left(x\right)=\frac{2{x}^{2}}{{x}^{2}-8x}[/latex] and [latex]g\left(x\right)=\frac{8{x}^{2}+24x}{{x}^{2}+x-6}[/latex].
Solution
[latex]R\left(x\right)=\frac{x-2}{4\left(x-8\right)}[/latex]
Try It
27) Find [latex]R\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}[/latex] where [latex]f\left(x\right)=\frac{15{x}^{2}}{3{x}^{2}+33x}[/latex] and [latex]g\left(x\right)=\frac{5x-5}{{x}^{2}+9x-22}[/latex].
Solution
[latex]R\left(x\right)=\frac{x\left(x-2\right)}{x-1}[/latex]
Key Concepts
- Determine the values for which a rational expression is undefined.
- Set the denominator equal to zero.
- Solve the equation.
- Equivalent Fractions Property
If [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] are numbers where [latex]b\ne 0,c\ne 0[/latex], then [latex]\frac{a}{b}=\frac{a·c}{b·c}[/latex] and [latex]\frac{a\cdot c}{b\cdot c}=\frac{a}{b}[/latex].
- How to simplify a rational expression.
- Factor the numerator and denominator completely.
- Simplify by dividing out common factors.
- Opposites in a Rational Expression
The opposite of [latex]a-b[/latex] is [latex]b-a[/latex]. [latex]\frac{a-b}{b-a}=-1\;a\ne b[/latex]
An expression and its opposite divide to [latex]-1[/latex].
- Multiplication of Rational Expressions
If [latex]p[/latex], [latex]q[/latex], [latex]r[/latex], and [latex]s[/latex] are polynomials where [latex]q\ne 0,s\ne 0[/latex], then
[latex]\frac{p}{q}\cdot\frac{r}{s}=\frac{pr}{qs}[/latex]
- How to multiply rational expressions.
- Factor each numerator and denominator completely.
- Multiply the numerators and denominators.
- Simplify by dividing out common factors.
- Division of Rational Expressions
If [latex]p[/latex], [latex]q[/latex], [latex]r[/latex], and [latex]s[/latex] are polynomials where [latex]q\ne 0,r\ne 0,s\ne 0[/latex], then
[latex]\frac{p}{q}\div\frac{r}{s}=\frac{p}{q}\cdot\frac{s}{r}[/latex]
- How to divide rational expressions.
- Rewrite the division as the product of the first rational expression and the reciprocal of the second.
- Factor the numerators and denominators completely.
- Multiply the numerators and denominators together.
- Simplify by dividing out common factors.
- How to determine the domain of a rational function.
- Set the denominator equal to zero.
- Solve the equation.
- The domain is all real numbers excluding the values found in Step 2:
Self Check
After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
Glossary
- rational expression
- A rational expression is an expression of the form [latex]\frac{p}{q},[/latex] where [latex]p[/latex] and [latex]q[/latex] are polynomials and [latex]q\ne 0.[/latex]
- simplified rational expression
- A simplified rational expression has no common factors, other than [latex]1[/latex], in its numerator and denominator.
- rational function
- A rational function is a function of the form [latex]R\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}[/latex] where [latex]p\left(x\right)[/latex] and [latex]q\left(x\right)[/latex] are polynomial functions and [latex]q\left(x\right)[/latex] is not zero.
A rational expression is an expression of the form [latex]\frac{p}{q},[/latex] where [latex]p[/latex] and [latex]q[/latex] are polynomials and [latex]q\ne 0.[/latex]
A simplified rational expression has no common factors, other than [latex]1[/latex], in its numerator and denominator.
A rational function is a function of the form [latex]R\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}[/latex] where [latex]p\left(x\right)[/latex] and [latex]q\left(x\right)[/latex] are polynomial functions and [latex]q\left(x\right)[/latex] is not zero.
