8.1 Simplify Rational Expressions

Learning objectives

Learning objectives

• Determine the values for which a rational expression is undefined

• Evaluate rational expressions

• Simplify rational expressions

• Simplify rational expressions with opposite factors

In Chapter 1, we reviewed the properties of fractions and their operations. We introduced rational numbers, which are just fractions where the numerators and denominators are integers, and the denominator is not zero.

In this chapter, we will work with fractions whose numerators and denominators are polynomials. We call these rational expressions.

Remember, division by 0 is undefined.

Here are some examples of rational expressions:

Notice that the first rational expression listed above, $-\frac{13}{42},$ 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 perform the same operations with rational expressions that we do with fractions. We will simplify, add, subtract, multiply, divide, and use them in applications.

Determine the Values for Which a Rational Expression is Undefined

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.

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

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.

Example 8.1

Solution

The expression will be undefined when the denominator is zero.

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Saying that the rational expression $\frac{x+4}{{x}^{2}+5x+6}$ is undefined for $x=-2\,\mathrm{or}\,x=-3$ is similar to writing the phrase “void where prohibited” in contest rules.

Evaluate Rational Expressions

To evaluate a rational expression, we substitute values of the variables into the expression and simplify, just as we have for many other expressions in this course.

Example 8.2

Solution

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Example 8.3

Solution

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Remember that a fraction is simplified when it has no common factors, other than 1, in its numerator and denominator. When we evaluate a rational expression, we make sure to simplify the resulting fraction.

Example 8.4

Solution

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Simplify Rational Expressions

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

For example:

• $\frac{2}{3}$ is simplified because there are no common factors of 2 and 3.

• $\frac{2x}{3x}$ is not simplified because x is a common factor of 2*x* and 3*x*.

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

Notice that in the Equivalent Fractions Property, the values that would make the denominators zero are specifically disallowed. We see $b\ne{}0,c\ne{}0$ clearly stated. Every time we write a rational expression, we should make a similar 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.

Let’s start by reviewing how we simplify numerical fractions.

Example 8.5

Solution

Notice that the fraction $-\frac{4}{7}$ is simplified because there are no more common factors.

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, $x\ne{}0$ and $y\ne{}0$.

Example 8.6

Solution

Did you notice that these are the same steps we took when we divided monomials in Polynomials?

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.

Note that removing the x’s from $\frac{x+5}{x}$ would be like cancelling the 2’s in the fraction $\frac{2+5}{2}$!

Example 8.7

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

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 covered in Factoring to factor the polynomials in the numerators and denominators in the following examples.

Example 8.8

Solution

$\begin{array}{cccc} & & & \,\frac{{x}^{2}+5x+6}{{x}^{2}+8x+12} \\ \mathrm{Factor the numerator and denominator.} & & & \,\frac{(x+2)(x+3)}{(x+2)(x+6)} \\ \begin{array}{l} \mathrm{Remove the common factor}\,x+2\,\mathrm{from} \\ \mathrm{the numerator and the denominator.} \\ \end{array} & & & \,\frac{(x+2)(x+3)}{(x+2)(x+6)} \\ & & & \,\frac{x+3}{x+6} \\ \end{array}$

Can you tell which values of x must be excluded in this example?

Example 8.9

Solution

$\begin{array}{cccc} & & & \,\frac{{y}^{2}+y-42}{{y}^{2}-36} \\ \mathrm{Factor the numerator and denominator.} & & & \,\frac{(y+7)(y-6)}{(y+6)(y-6)} \\ \begin{array}{l} \mathrm{Remove the common factor}\,y-6\,\mathrm{from} \\ \mathrm{the numerator and the denominator.} \\ \end{array} & & & \,\frac{(y+7)(y-6)}{(y+6)(y-6)} \\ & & & \,\frac{y+7}{y+6} \\ \end{array}$

Example 8.10

Solution

$\begin{array}{cccc} & & & \,\frac{{p}^{3}-2{p}^{2}+2p-4}{{p}^{2}-7p+10} \\ \begin{array}{l} \mathrm{Factor the numerator and denominator,} \\ \mathrm{using grouping to factor the numerator.} \\ \end{array} & & & \,\frac{{p}^{2}(p-2)+2(p-2)}{(p-5)(p-2)} \\ & & & \,\frac{({p}^{2}+2)(p-2)}{(p-5)(p-2)} \\ \begin{array}{l} \mathrm{Remove the common factor of}\,p-2 \\ \mathrm{from the numerator and the denominator.} \\ \end{array} & & & \,\frac{({p}^{2}+2)(p-2)}{(p-5)(p-2)} \\ & & & \,\frac{{p}^{2}+2}{p-5} \\ \end{array}$

Example 8.11

Solution

$\begin{array}{cccc} & & & \,\frac{2{n}^{2}-14n}{4{n}^{2}-16n-48} \\ \begin{array}{l} \mathrm{Factor the numerator and denominator,} \\ \mathrm{first factoring out the GCF.} \\ \end{array} & & & \,\frac{2n(n-7)}{4({n}^{2}-4n-12)} \\ & & & \,\frac{2n(n-7)}{4(n-6)(n+2)} \\ \mathrm{Remove the common factor, 2.} & & & \,\frac{2n(n-7)}{2\cdot{}2(n-6)(n+2)} \\ & & & \,\frac{n(n-7)}{2(n-6)(n+2)} \\ \end{array}$

Example 8.12

Solution

$\begin{array}{cccc} & & & \,\frac{3{b}^{2}-12b+12}{6{b}^{2}-24} \\ \begin{array}{l} \mathrm{Factor the numerator and denominator,} \\ \mathrm{first factoring out the GCF.} \\ \end{array} & & & \,\frac{3({b}^{2}-4b+4)}{6({b}^{2}-4)} \\ & & & \,\frac{3(b-2)(b-2)}{6(b+2)(b-2)} \\ \mathrm{Remove the common factors of}\,b-2\,\mathrm{and}\,3. & & & \,\frac{3(b-2)(b-2)}{3\cdot{}2(b+2)(b-2)} \\ & & & \,\frac{b-2}{2(b+2)} \\ \end{array}$

Example 8.13

Solution

$\begin{array}{cccc} & & & \,\frac{{m}^{3}+8}{{m}^{2}-4} \\ \begin{array}{l} \mathrm{Factor the numerator and denominator,} \\ \mathrm{using the formulas for sum of cubes and} \\ \mathrm{difference of squares.} \\ \end{array} & & & \,\frac{(m+2)({m}^{2}-2m+4)}{(m+2)(m-2)} \\ \mathrm{Remove the common factor of}\,m+2. & & & \,\frac{(m+2)({m}^{2}-2m+4)}{(m+2)(m-2)} \\ & & & \,\frac{{m}^{2}-2m+4}{m-2} \\ \end{array}$

Simplify Rational Expressions with Opposite Factors

Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors. Let’s start with a numerical fraction, say $\frac{7}{-7}$. We know this fraction simplifies to $-1$. We also recognize that the numerator and denominator are opposites.

In Foundations, we introduced opposite notation: the opposite of $a$ is $-a$. We remember, too, that $-a=-1\cdot{}a$.

We simplify the fraction $\frac{a}{-a}$, whose numerator and denominator are opposites, in this way:

$\begin{array}{cccc} & & & \,\frac{a}{-a} \\ \mathrm{We could rewrite this.} & & & \,\frac{1\cdot{}a}{-1\cdot{}a} \\ \mathrm{Remove the common factors.} & & & \,\frac{1}{-1} \\ \mathrm{Simplify.} & & & \,-1 \\ \end{array}$

So, in the same way, we can simplify the fraction $\frac{x-3}{-(x-3)}$:

$\begin{array}{lccc} \mathrm{We could rewrite this.} & & & \,\frac{1\cdot(x-3)}{-1\cdot(x-3)} \\ \mathrm{Remove the common factors.} & & & \,\frac{1}{-1} \\ \mathrm{Simplify.} & & & \,-1 \\ \end{array}$

But the opposite of $x-3$ could be written differently:

$\begin{array}{cccc} & & & \,-(x-3) \\ \mathrm{Distribute.} & & & \,-x+3 \\ \mathrm{Rewrite.} & & & \,3-x \\ \end{array}$

This means the fraction $\frac{x-3}{3-x}$ simplifies to $-1$.

In general, we could write the opposite of $a-b$ as $b-a$. So the rational expression $\frac{a-b}{b-a}$ simplifies to $-1$.

We will use this property to simplify rational expressions that contain opposites in their numerators and denominators.

Example 8.14

Solution

$\begin{array}{cccl} & & & \frac{x-8}{8-x} \\ \mathrm{Recognize that}\,x-8\,\mathrm{and}\,8-x\,\mathrm{are opposites.} & & & -1 \\ \end{array}$

Remember, the first step in simplifying a rational expression is to factor the numerator and denominator completely.

Example 8.15

Solution

Example 8.16

Solution

Section 8.1 Exercises

Section 8.1 Exercises

In the following exercises, determine the values for which the rational expression is undefined.

Evaluate Rational Expressions

In the following exercises, evaluate the rational expression for the given values.

Simplify Rational Expressions

In the following exercises, simplify.

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify each rational expression.

Everyday Math

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential - every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.

1. Determine the values for which the rational expression is undefined:

ⓐ $\frac{3y}{x}$ ⓑ $\frac{8n-5}{3n+1}$ ⓒ $\frac{a+10}{{a}^{2}+4a+3}$

2. Determine the values for which the rational expression is undefined:

ⓐ $\frac{4p}{5q}$ ⓑ $\frac{y-1}{3y+2}$ ⓒ $\frac{m-5}{{m}^{2}+m-6}$

3. Evaluate $\frac{y+1}{2y-3}$ for each value:

ⓐ $y=1$ ⓑ $y=-3$ ⓒ $y=0$

4. Evaluate $\frac{5x-1}{2x+1}$ for each value:

ⓐ $x=1$ ⓑ $x=-1$ ⓒ $x=0$

5. Evaluate $\frac{{x}^{2}+1}{{x}^{2}-3x+2}$ for each value:

ⓐ $x=0$ ⓑ $x=-1$ ⓒ $x=3$

1. Evaluate $\frac{{x}^{2}+x-6}{{x}^{2}-9}$ for each value:

ⓐ $x=0$ ⓑ $x=-2$ ⓒ $x=1$

2. Evaluate $\frac{2{a}^{3}b}{{a}^{2}+2ab+{b}^{2}}$ for each value:

ⓐ $a=-1,b=2$ ⓑ $a=0,b=-1$ ⓒ $a=1,b=\frac{1}{2}$

3. Evaluate $\frac{{a}^{2}-{b}^{2}}{8a{b}^{3}}$ for each value:

ⓐ $a=1,b=-1$ ⓑ $a=\frac{1}{2},b=-1$ ⓒ $a=-2,b=1$

4. Simplify: $-\frac{45}{81}.$

5. Simplify: $-\frac{42}{54}.$

1. Simplify: $\frac{4{x}^{2}y}{12x{y}^{2}}.$

2. Simplify: $\frac{16{x}^{2}y}{2x{y}^{2}}.$

3. Simplify: $\frac{3x-6}{2x-4}.$

4. Simplify: $\frac{7y+35}{5y+25}.$

5. Simplify: $\frac{{x}^{2}-x-2}{{x}^{2}-3x+2}.$

1. Simplify: $\frac{{x}^{2}-3x-10}{{x}^{2}+x-2}.$

2. Simplify: $\frac{{x}^{2}+x-6}{{x}^{2}-4}.$

3. Simplify: $\frac{{x}^{2}+8x+7}{{x}^{2}-49}.$

4. Simplify: $\frac{{y}^{3}-3{y}^{2}+y-3}{{y}^{2}-y-6}.$

5. Simplify: $\frac{{p}^{3}-{p}^{2}+2p-2}{{p}^{2}+4p-5}.$

1. Simplify: $\frac{2{n}^{2}-10n}{4{n}^{2}-16n-20}.$

2. Simplify: $\frac{4{x}^{2}-16x}{8{x}^{2}-16x-64}.$

3. Simplify: $\frac{2{x}^{2}-12x+18}{3{x}^{2}-27}.$

4. Simplify: $\frac{5{y}^{2}-30y+25}{2{y}^{2}-50}.$

5. Simplify: $\frac{{p}^{3}-64}{{p}^{2}-16}.$

1. Simplify: $\frac{{x}^{3}+8}{{x}^{2}-4}.$

2. Simplify: $\frac{y-2}{2-y}.$

3. Simplify: $\frac{n-9}{9-n}.$

4. Simplify: $\frac{10-2y}{{y}^{2}-25}.$

5. Simplify: $\frac{3y-27}{81-{y}^{2}}.$

1. Simplify: $\frac{{x}^{2}-4x-5}{25-{x}^{2}}.$

2. Simplify: $\frac{{x}^{2}+x-2}{1-{x}^{2}}.$

3. ⓐ $\frac{2x}{z}$ ⓑ $\frac{4p-1}{6p-5}$ ⓒ $\frac{n-3}{{n}^{2}+2n-8}$

4. ⓐ $\frac{4{x}^{2}y}{3y}$ ⓑ $\frac{3x-2}{2x+1}$ ⓒ $\frac{u-1}{{u}^{2}-3u-28}$

5. $\frac{2x}{x-1}$

ⓐ $x=0$ ⓑ $x=2$ ⓒ $x=-1$

1. $\frac{2p+3}{{p}^{2}+1}$

ⓐ $p=0$ ⓑ $p=1$ ⓒ $p=-2$

2. $\frac{{y}^{2}+5y+6}{{y}^{2}-1}$

ⓐ $y=0$ ⓑ $y=2$ ⓒ $y=-2$

3. $\frac{{a}^{2}-4}{{a}^{2}+5a+4}$

ⓐ $a=0$ ⓑ $a=1$ ⓒ $a=-2$

4. $\frac{{x}^{2}+3xy+2{y}^{2}}{2{x}^{3}y}$

1. ⓐ $x=1,y=-1$

2. ⓑ $x=2,y=1$

3. ⓒ $x=-1,y=-2$

5. $\frac{{m}^{2}-4{n}^{2}}{5m{n}^{3}}$

1. ⓐ $m=2,n=1$

2. ⓑ $m=-1,n=-1$

3. ⓒ $m=3,n=2$

1. $-\frac{4}{52}$

2. $\frac{56}{63}$

3. $\frac{6a{b}^{2}}{12{a}^{2}b}$

4. $\frac{8{m}^{3}n}{12m{n}^{2}}$

5. $\frac{3a+6}{4a+8}$

1. $\frac{3c-9}{5c-15}$

2. $\frac{7m+63}{5m+45}$

3. $\frac{12p-240}{5p-100}$

4. $\frac{{a}^{2}-a-12}{{a}^{2}-8a+16}$

5. $\frac{{y}^{2}+3y-4}{{y}^{2}-6y+5}$

1. $\frac{{x}^{2}-25}{{x}^{2}+2x-15}$

2. $\frac{{y}^{2}-2y-3}{{y}^{2}-9}$

3. $\frac{{y}^{3}+{y}^{2}+y+1}{{y}^{2}+2y+1}$

4. $\frac{{x}^{3}-2{x}^{2}-25x+50}{{x}^{2}-25}$

5. $\frac{3{a}^{2}+15a}{6{a}^{2}+6a-36}$

1. $\frac{-5{c}^{2}-10c}{-10{c}^{2}+30c+100}$

2. $\frac{3{m}^{2}+30m+75}{4{m}^{2}-100}$

3. $\frac{5{r}^{2}+30r-35}{{r}^{2}-49}$

4. $\frac{{t}^{3}-27}{{t}^{2}-9}$

5. $\frac{{w}^{3}+216}{{w}^{2}-36}$

1. $\frac{a-5}{5-a}$

2. $\frac{11-c}{c-11}$

3. $\frac{12-2x}{{x}^{2}-36}$

4. $\frac{4v-32}{64-{v}^{2}}$

5. $\frac{{y}^{2}-11y+24}{9-{y}^{2}}$

1. $\frac{{a}^{2}-5a-36}{81-{a}^{2}}$

2. Tax Rates For the tax year 2015, the amount of tax owed by a single person earning between $37,450 and$90,750, can be found by evaluating the formula $0.25x-4206.25,$ where x is income. The average tax rate for this income can be found by evaluating the formula $\frac{0.25x-4206.25}{x}.$ What would be the average tax rate for a single person earning $50,000?

3. Explain how you find the values of x for which the rational expression $\frac{{x}^{2}-x-20}{{x}^{2}-4}$ is undefined.