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# Addition and Subtraction of Rational Expressions

## Add and subtract fractions with variables in the denominator

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Addition and Subtraction of Rational Expressions

### Addition and Subtraction of Rational Expressions

Like fractions, rational expressions represent a portion of a quantity. Remember that when we add or subtract fractions we must first make sure that they have the same denominator. Once the fractions have the same denominator, we combine the different portions by adding or subtracting the numerators and writing that answer over the common denominator.

#### Add and Subtract Rational Expressions with the Same Denominator

Fractions with common denominators combine in the following manner:

ac+bc=a+bcandacbc=abc\begin{align*}\frac{a}{c}+\frac{b}{c} = \frac{a+b}{c} \qquad \text{and} \qquad \frac{a}{c} - \frac{b}{c}=\frac{a-b}{c}\end{align*}

#### Simplifying Expressions

Simplify.

a) 8727+47\begin{align*}\frac{8}{7} - \frac{2}{7} + \frac{4}{7}\end{align*}

Since the denominators are the same we combine the numerators:

8727+47=82+47=107\begin{align*}\frac{8}{7} - \frac{2}{7} + \frac{4}{7} = \frac{8-2+4}{7} = \frac{10}{7}\end{align*}

b) 4x23x+5+2x21x+5\begin{align*}\frac{4x^2-3}{x+5} + \frac{2x^2-1}{x+5}\end{align*}

Since the denominators are the same we combine the numerators:4x23+2x21x+5Simplify by collecting like terms:6x24x+5\begin{align*}\text{Since the denominators are the same we combine the numerators:} \qquad \frac{4x^2-3+2x^2-1}{x+5}\! \\ \text{Simplify by collecting like terms:} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \frac{6x^2-4}{x+5}\end{align*}

c) x22x+12x+33x23x+52x+3\begin{align*}\frac{x^2-2x+1}{2x+3} - \frac{3x^2-3x+5}{2x+3}\end{align*}

Since the denominators are the same we combine the numerators. Make sure the subtraction sign is distributed to all terms in the second expression:

x22x+1(3x23x+5)2x+3=x22x+13x2+3x52x+3=2x2+x42x+3\begin{align*}\frac{x^2-2x+1-(3x^2-3x+5)}{2x+3} = \frac{x^2-2x+1-3x^2+3x-5}{2x+3}= \frac{-2x^2+x-4}{2x+3}\end{align*}

#### Finding the Least Common Denominator of Rational Expressions

To add and subtract fractions with different denominators, we must first rewrite all fractions so that they have the same denominator. In general, we want to find the least common denominator. To find the least common denominator, we find the least common multiple (LCM) of the expressions in the denominators of the different fractions. Remember that the least common multiple of two or more integers is the least positive integer that has all of those integers as factors.

The procedure for finding the lowest common multiple of polynomials is similar. We rewrite each polynomial in factored form and we form the LCM by taking each factor to the highest power it appears in any of the separate expressions.

1. Find the LCM of 48x2y\begin{align*}48x^2y\end{align*} and 60xy3z\begin{align*}60xy^3z\end{align*}.

First rewrite the integers in their prime factorization.

4860=243=2235\begin{align*}48 & = 2^4 \cdot 3\\ 60 & = 2^2 \cdot 3 \cdot 5\end{align*}

The two expressions can be written as:

48x2y=243x2y60xy3z=2235xy3z\begin{align*}& 48x^2y=2^4 \cdot 3 \cdot x^2 \cdot y\\ & 60xy^3z=2^2 \cdot 3 \cdot 5 \cdot x \cdot y^3 \cdot z\end{align*}

To find the LCM, take the highest power of each factor that appears in either expression.

LCM=2435x2y3z=240x2y3z\begin{align*}\text{LCM} = 2^4 \cdot 3 \cdot 5 \cdot x^2 \cdot y^3 \cdot z = 240x^2y^3z\end{align*}

2. Find the LCM of 2x2+8x+8\begin{align*}2x^2+8x+8\end{align*} and x34x212x\begin{align*}x^3-4x^2-12x\end{align*}

Factor the polynomials completely:

2x2+8x+8=2(x2+4x+4)=2(x+2)2\begin{align*}2x^2+8x+8 & = 2(x^2+4x+4)\\ & = 2(x+2)^2\end{align*}

x34x212x=x(x24x12)=x(x+2)(x6)\begin{align*}x^3-4x^2-12x & = x(x^2-4x-12)\\ & = x(x+2)(x-6)\end{align*}

To find the LCM, take the highest power of each factor that appears in either expression.

LCM=2x(x+2)2(x6)\begin{align*}\text{LCM} = 2x(x+2)^2 (x-6)\end{align*}

It’s customary to leave the LCM in factored form, because this form is useful in simplifying rational expressions and finding any excluded values.\

#### Add and Subtract Rational Expressions with Different Denominators

Now we’re ready to add and subtract rational expressions. We use the following procedure.

1. Find the least common denominator (LCD) of the fractions.
2. Express each fraction as an equivalent fraction with the LCD as the denominator.
3. Add or subtract and simplify the result.

1. Perform the following operation and simplify: 2x+232x5\begin{align*}\frac{2}{x+2} - \frac{3}{2x-5}\end{align*}

The denominators can’t be factored any further, so the LCD is just the product of the separate denominators: (x+2)(2x5)\begin{align*}(x+2)(2x-5)\end{align*}. That means the first fraction needs to be multiplied by the factor (2x5)\begin{align*}(2x-5)\end{align*} and the second fraction needs to be multiplied by the factor (x+2)\begin{align*}(x+2)\end{align*}:

2x+2(2x5)(2x5)32x5(x+2)(x+2)\begin{align*}\frac{2}{x+2} \cdot \frac{(2x-5)}{(2x-5)} - \frac{3}{2x-5} \cdot \frac{(x+2)}{(x+2)}\end{align*}

\begin{align*}\text{Combine the numerators and simplify:} \qquad \qquad \frac{2(2x-5)-3(x+2)}{(x+2)(2x-5)} = \frac{4x-10-3x-6}{(x+2)(2x-5)}\!\\ \\ \text{Combine like terms in the numerator:} \qquad \qquad \frac{x-16}{(x+2)(2x-5)} \quad \mathbf{Answer}\end{align*}

2. Perform the following operation and simplify: \begin{align*}\frac{4x}{x-5}-\frac{3x}{5-x}\end{align*}.

Notice that the denominators are almost the same; they just differ by a factor of -1.\begin{align*}\text{Factor out -1 from the second denominator:} \qquad \qquad \qquad \qquad \qquad \qquad \frac{4x}{x-5} - \frac{3x}{-(x-5)}\!\\ \\ \text{The two negative signs in the second fraction cancel:} \qquad \qquad \qquad \qquad \frac{4x}{x-5}+\frac{3x}{(x-5)}\!\\ \\ \text{Since the denominators are the same we combine the numerators:} \ \qquad \frac{7x}{x-5} \quad \mathbf{Answer}\end{align*}

### Examples

#### Example 1

Find the LCM of \begin{align*}x^2-25\end{align*} and \begin{align*}x^2+3x+2\end{align*}.

First factor each polynomial to see if they have any common factors:

\begin{align*}x^2-25=(x+5)(x-5)\end{align*} and \begin{align*}x^2+3x+2=(x+2)(x+1)\end{align*}

Since the two polynomials do not have any common factors, this means that the LCM of the two polynomials is:

\begin{align*} (x^2-25)(x^2+3x+2)=x^4+3x^3-23x^2-75x-50\end{align*}

#### Example 2

Perform the following operation and simplify: \begin{align*}\frac{2x-1}{x^2-9}-\frac{3x+4}{x^2-9}\end{align*}.

To subtract the second fraction from the first, subtraction the numerator of the second from the numerator of the first. Make sure to put parenthesis around the numerator of the second fraction, so you remember to subtract each term.

\begin{align*}\frac{2x-1}{x^2-9}-\frac{3x+4}{x^2-9}=\frac{2x-1-(3x+4)}{x^2-9}=\frac{2x-1-3x-4}{x^2-9}=\frac{-x-5}{x^2-9}\end{align*}

### Review

Perform the indicated operation and simplify. Leave the denominator in factored form.

1. \begin{align*}\frac{5}{24}-\frac{7}{24}\end{align*}
2. \begin{align*}\frac{2x}{13}-\frac{x}{3}\end{align*}
3. \begin{align*}\frac{5}{2x+3}+\frac{3}{2x+3}\end{align*}
4. \begin{align*}\frac{1}{5x-7}+\frac{10}{5x-7}\end{align*}
5. \begin{align*}\frac{3x-1}{x+9}-\frac{4x+3}{x+9}\end{align*}
6. \begin{align*}\frac{1-7x}{3x+10}-\frac{x+20}{3x+10}\end{align*}
7. \begin{align*}\frac{4x+7}{2x^2}-\frac{3x-4}{2x^2}\end{align*}
8. \begin{align*}\frac{10x-5}{9x^2}-\frac{5}{9x^2}\end{align*}
9. \begin{align*}\frac{x^2}{x+5}-\frac{25}{x+5}\end{align*}
10. \begin{align*}\frac{.25x^2}{x+100}-\frac{0.1}{x+100}\end{align*}
11. \begin{align*}\frac{1}{x}+\frac{2}{3x}\end{align*}
12. \begin{align*}\frac{4}{5x^2}-\frac{2}{7x^3}\end{align*}
13. \begin{align*}\frac{10}{3x-1}-\frac{7}{1-3x}\end{align*}
14. \begin{align*}\frac{10}{x+5}+\frac{2}{x+2}\end{align*}
15. \begin{align*}\frac{2x}{x-3}-\frac{3x}{x+4}\end{align*}
16. \begin{align*}\frac{4x-3}{2x+1}+\frac{x+2}{x-9}\end{align*}
17. \begin{align*}\frac{x^2}{x+4}-\frac{3x^2}{4x-1}\end{align*}
18. \begin{align*}\frac{2}{5x+2}-\frac{x+1}{x^2}\end{align*}
19. \begin{align*}\frac{x+4}{2x}+\frac{2}{9x}\end{align*}
20. \begin{align*}\frac{5x+3}{x^2+x}+\frac{2x+1}{x}\end{align*}
21. \begin{align*}\frac{4}{(x+1)(x-1)}-\frac{5}{(x+1)(x+2)}\end{align*}
22. \begin{align*}\frac{2x}{(x+2)(3x-4)}+\frac{7x}{(3x-4)^2}\end{align*}
23. \begin{align*}\frac{3x+5}{x(x-1)}-\frac{9x-1}{(x-1)^2}\end{align*}
24. \begin{align*}\frac{1}{(x-2)(x-3)}+\frac{4}{(2x+5)(x-6)}\end{align*}
25. \begin{align*}\frac{3x-2}{x-2}+\frac{1}{x^2-4x+4}\end{align*}
26. \begin{align*}\frac{-x^3}{x^2-7x+6}+x-4\end{align*}
27. \begin{align*}\frac{2x}{x^2+10x+25}-\frac{3x}{2x^2+7x-15}\end{align*}
28. \begin{align*}\frac{1}{x^2-9}+\frac{2}{x^2+5x+6}\end{align*}
29. \begin{align*}\frac{-x+4}{2x^2-x-15}+\frac{x}{4x^2+8x-5}\end{align*}
30. \begin{align*}\frac{4}{9x^2-49}-\frac{1}{3x^2+5x-28}\end{align*}

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### Vocabulary Language: English

Least Common Denominator

The least common denominator or lowest common denominator of two fractions is the smallest number that is a multiple of both of the original denominators.