7.2 Add and Subtract Rational Expressions

Learning objectives

Learning objectives

• Add and subtract rational expressions with a common denominator

• Add and subtract rational expressions whose denominators are opposites

• Find the least common denominator of rational expressions

• Add and subtract rational expressions with unlike denominators

• Add and subtract rational functions

Add and Subtract Rational Expressions with a Common Denominator

What is the first step you take when you add numerical fractions? You check if they have a common denominator. If they do, you add the numerators and place the sum over the common denominator. If they do not have a common denominator, you find one before you add.

It is the same with rational expressions. To add rational expressions, they must have a common denominator. When the denominators are the same, you add the numerators and place the sum over the common denominator.

To add or subtract rational expressions with a common denominator, add or subtract the numerators and place the result over the common denominator.

We always simplify rational expressions. Be sure to factor, if possible, after you subtract the numerators so you can identify any common factors.

Remember, too, we do not allow values that would make the denominator zero. What value of x should be excluded in the next example?

Example 7.13

Solution

Since the denominator is $x+4,$ we must exclude the value $x=-4.$

$\,\frac{11x+28}{x+4}+\frac{{x}^{2}}{x+4},\,x\ne-4$

The fractions have a common denominator,so add the numerators and place the sumover the common denominator. · $\,\frac{11x+28+{x}^{2}}{x+4}$ Write the degrees in descending order. · $\,\frac{{x}^{2}+11x+28}{x+4}$ Factor the numerator. · $\,\frac{(x+4)(x+7)}{x+4}$ Simplify by removing common factors. · $\,\frac{(x+4)(x+7)}{x+4}$ Simplify. · $\,x+7$

The expression simplifies to $x+7$ but the original expression had a denominator of $x+4$ so $x\ne-4.$

To subtract rational expressions, they must also have a common denominator. When the denominators are the same, you subtract the numerators and place the difference over the common denominator. Be careful of the signs when you subtract a binomial or trinomial.

Example 7.14

Solution

$\,\frac{5{x}^{2}-7x+3}{{x}^{2}-3x-18}-\frac{4{x}^{2}+x-9}{{x}^{2}-3x-18}$

Subtract the numerators and place thedifference over the common denominator. · $\,\frac{5{x}^{2}-7x+3-(4{x}^{2}+x-9)}{{x}^{2}-3x-18}$ Distribute the sign in the numerator. · $\,\frac{5{x}^{2}-7x+3-4{x}^{2}-x+9}{{x}^{2}-3x-18}$ Combine like terms. · $\,\frac{{x}^{2}-8x+12}{{x}^{2}-3x-18}$ Factor the numerator and the denominator. · $\,\frac{(x-2)(x-6)}{(x+3)(x-6)}$ Simplify by removing common factors. · $\,\frac{(x-2)(x-6)}{(x+3)(x-6)}$

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

Add and Subtract Rational Expressions Whose Denominators are Opposites

When the denominators of two rational expressions are opposites, it is easy to get a common denominator. We just have to multiply one of the fractions by $\frac{-1}{-1}.$

Let’s see how this works.

Be careful with the signs as you work with the opposites when the fractions are being subtracted.

Example 7.15

Solution

Find the Least Common Denominator of Rational Expressions

When we add or subtract rational expressions with unlike denominators, we will need to get common denominators. If we review the procedure we used with numerical fractions, we will know what to do with rational expressions.

Let’s look at this example: $\frac{7}{12}+\frac{5}{18}.$ Since the denominators are not the same, the first step was to find the least common denominator (LCD).

To find the LCD of the fractions, we factored 12 and 18 into primes, lining up any common primes in columns. Then we “brought down” one prime from each column. Finally, we multiplied the factors to find the LCD.

When we add numerical fractions, once we found the LCD, we rewrote each fraction as an equivalent fraction with the LCD by multiplying the numerator and denominator by the same number. We are now ready to add.

We do the same thing for rational expressions. However, we leave the LCD in factored form.

Remember, we always exclude values that would make the denominator zero. What values of $x$ should we exclude in this next example?

Example 7.16

Solution

ⓐ

ⓑ

Add and Subtract Rational Expressions with Unlike Denominators

Now we have all the steps we need to add or subtract rational expressions with unlike denominators.

Example 7.17

The steps used to add rational expressions are summarized here.

Avoid the temptation to simplify too soon. In the example above, we must leave the first rational expression as $\frac{3x-6}{(x-3)(x-2)}$ to be able to add it to $\frac{2x-6}{(x-2)(x-3)}.$ Simplify only after you have combined the numerators.

Example 7.18

Solution

The process we use to subtract rational expressions with different denominators is the same as for addition. We just have to be very careful of the signs when subtracting the numerators.

Example 7.19

Solution

There are lots of negative signs in the next example. Be extra careful.

Example 7.20

Solution

Things can get very messy when both fractions must be multiplied by a binomial to get the common denominator.

Example 7.21

Solution

We follow the same steps as before to find the LCD when we have more than two rational expressions. In the next example, we will start by factoring all three denominators to find their LCD.

Example 7.22

Solution

Add and subtract rational functions

To add or subtract rational functions, we use the same techniques we used to add or subtract polynomial functions.

Example 7.23

Solution

Section 7.2 Exercises

Section 7.2 Exercises

Add and Subtract Rational Expressions with a Common Denominator

In the following exercises, add.

In the following exercises, subtract.

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add or subtract.

Find the Least Common Denominator of Rational Expressions

In the following exercises, ⓐ find the LCD for the given rational expressions ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

Add and Subtract Rational Expressions with Unlike Denominators

In the following exercises, perform the indicated operations.

Add and Subtract Rational Functions

In the following exercises, find ⓐ $R(x)=f(x)+g(x)$ ⓑ $R(x)=f(x)-g(x).$

Writing Exercises

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

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

1. Simplify: $\frac{9x+14}{x+7}+\frac{{x}^{2}}{x+7}.$

2. Simplify: $\frac{{x}^{2}+8x}{x+5}+\frac{15}{x+5}.$

3. Subtract: $\frac{4{x}^{2}-11x+8}{{x}^{2}-3x+2}-\frac{3{x}^{2}+x-3}{{x}^{2}-3x+2}.$

4. Subtract: $\frac{6{x}^{2}-x+20}{{x}^{2}-81}-\frac{5{x}^{2}+11x-7}{{x}^{2}-81}.$

1. Subtract: $\frac{{y}^{2}-5y}{{y}^{2}-4}-\frac{6y-6}{4-{y}^{2}}.$

2. Subtract: $\frac{2{n}^{2}+8n-1}{{n}^{2}-1}-\frac{{n}^{2}-7n-1}{1-{n}^{2}}.$

3. ⓐ Find the LCD for the expressions $\frac{2}{{x}^{2}-x-12},\frac{1}{{x}^{2}-16}$ ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

4. ⓐ Find the LCD for the expressions $\frac{3x}{{x}^{2}-3x-10},\frac{5}{{x}^{2}+3x+2}$ ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.

1. Add: $\frac{2}{x-2}+\frac{5}{x+3}.$

2. Add:$\frac{4}{m+3}+\frac{3}{m+4}.$

3. Add: $\frac{1}{{m}^{2}-m-2}+\frac{5m}{{m}^{2}+3m+2}.$

4. Add:$\frac{2n}{{n}^{2}-3n-10}+\frac{6}{{n}^{2}+5n+6}.$

1. Subtract: $\frac{2x}{{x}^{2}-4}-\frac{1}{x+2}.$

2. Subtract: $\frac{3}{z+3}-\frac{6z}{{z}^{2}-9}.$

3. Subtract :$\frac{3x-1}{{x}^{2}-5x-6}-\frac{2}{6-x}.$

4. Subtract: $\frac{-2y-2}{{y}^{2}+2y-8}-\frac{y-1}{2-y}.$

1. Subtract: $\frac{3}{{b}^{2}-4b-5}-\frac{2}{{b}^{2}-6b+5}.$

2. Subtract: $\frac{4}{{x}^{2}-4}-\frac{3}{{x}^{2}-x-2}.$

3. Simplify: $\frac{v}{v+1}+\frac{3}{v-1}-\frac{6}{{v}^{2}-1}.$

4. Simplify: $\frac{3w}{w+2}+\frac{2}{w+7}-\frac{17w+4}{{w}^{2}+9w+14}.$

1. Find $R(x)=f(x)-g(x)$ where $f(x)=\frac{x+1}{x+3}$ and $g(x)=\frac{x+17}{{x}^{2}-x-12}.$

2. Find $R(x)=f(x)+g(x)$ where $f(x)=\frac{x-4}{x+3}$ and $g(x)=\frac{4x+6}{{x}^{2}-9}.$

3. $\frac{2}{15}+\frac{7}{15}$

4. $\frac{3c}{4c-5}+\frac{5}{4c-5}$

1. $\frac{2{r}^{2}}{2r-1}+\frac{15r-8}{2r-1}$

2. $\frac{2{w}^{2}}{{w}^{2}-16}+\frac{8w}{{w}^{2}-16}$

3. $\frac{9{a}^{2}}{3a-7}-\frac{49}{3a-7}$

4. $\frac{3{m}^{2}}{6m-30}-\frac{21m-30}{6m-30}$

1. $\frac{6{p}^{2}+3p+4}{{p}^{2}+4p-5}-\frac{5{p}^{2}+p+7}{{p}^{2}+4p-5}$

2. $\frac{5{r}^{2}+7r-33}{{r}^{2}-49}-\frac{4{r}^{2}+5r+30}{{r}^{2}-49}$

3. $\frac{10v}{2v-1}+\frac{2v+4}{1-2v}$

4. $\frac{10{x}^{2}+16x-7}{8x-3}+\frac{2{x}^{2}+3x-1}{3-8x}$

1. $\frac{{z}^{2}+6z}{{z}^{2}-25}-\frac{3z+20}{25-{z}^{2}}$

2. $\frac{2{b}^{2}+30b-13}{{b}^{2}-49}-\frac{2{b}^{2}-5b-8}{49-{b}^{2}}$

3. $\frac{5}{{x}^{2}-2x-8},\frac{2x}{{x}^{2}-x-12}$

4. $\frac{9}{{z}^{2}+2z-8},\frac{4z}{{z}^{2}-4}$

1. $\frac{4}{{b}^{2}+6b+9},\frac{2b}{{b}^{2}-2b-15}$

2. $\frac{2}{3{d}^{2}+14d-5},\frac{5d}{3{d}^{2}-19d+6}$

3. $\frac{7}{10{x}^{2}y}+\frac{4}{15x{y}^{2}}$

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

1. $\frac{5}{3w-2}+\frac{2}{w+1}$

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

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

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

1. $\frac{3r}{{r}^{2}+7r+6}+\frac{9}{{r}^{2}+4r+3}$

2. $\frac{t}{t-6}-\frac{t-2}{t+6}$

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

4. $\frac{6}{m+6}-\frac{12m}{{m}^{2}-36}$

1. $\frac{-9p-17}{{p}^{2}-4p-21}-\frac{p+1}{7-p}$

2. $\frac{-2r-16}{{r}^{2}+6r-16}-\frac{5}{2-r}$

3. $\frac{2x+7}{10x-1}+3$

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

1. $\frac{5}{{x}^{2}+8x-9}-\frac{4}{{x}^{2}+10x+9}$

2. $\frac{5a}{a-2}+\frac{9}{a}-\frac{2a+18}{{a}^{2}-2a}$

3. $\frac{c}{c+2}+\frac{5}{c-2}-\frac{10c}{{c}^{2}-4}$

4. $\frac{3d}{d+2}+\frac{4}{d}-\frac{d+8}{{d}^{2}+2d}$

1. $f(x)=\frac{-5x-5}{{x}^{2}+x-6}$ and$\,g(x)=\frac{x+1}{2-x}$

2. $f(x)=\frac{6x}{{x}^{2}-64}$ and$\,g(x)=\frac{3}{x-8}$

3. Donald thinks that $\frac{3}{x}+\frac{4}{x}$ is $\frac{7}{2x}.$ Is Donald correct? Explain.

4. Felipe thinks $\frac{1}{x}+\frac{1}{y}$ is $\frac{2}{x+y}.$ ⓐ Choose numerical values for x and y and evaluate $\frac{1}{x}+\frac{1}{y}.$ ⓑ Evaluate $\frac{2}{x+y}$ for the same values of x and y you used in part ⓐ. ⓒ Explain why Felipe is wrong. ⓓ Find the correct expression for $\frac{1}{x}+\frac{1}{y}.$