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Addition and Subtraction of Rational Expressions

Add and subtract fractions with variables in the denominator

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Rational Expression Addition and Subtraction

Can you use your knowledge of rational expressions and adding fractions to add the following rational expressions?

\begin{align*}\frac{3x}{x^2+6x-16}+\frac{2x}{x-2}\end{align*}

Adding and Subtracting Rational Expressions

Rational expressions are examples of fractions, so you add and subtract rational expressions in the same way that you add and subtract fractions. As with fractions, you will need a common denominator, ideally the lowest common denominator (LCD), in order to add or subtract the expressions.

For example, to add:

\begin{align*}\frac{x^2+2x}{x+3}+ \frac{x}{x^2+4x+3}\end{align*}

First, factor the denominators to get:

\begin{align*}\frac{x^2+2x}{x+3}+ \frac{x}{(x+3)(x+1)}\end{align*}.

Next, find the lowest common denominator (LCD). This will be the product of each unique factor in the denominators. In this case, the LCD is \begin{align*}(x+3)(x+1)\end{align*}. Multiply the numerator and denominator of each fraction by the factors necessary to create the common denominator. In this case, you only need to multiply the fraction on the left by \begin{align*}\left(\frac{x+1}{x+1}\right)\end{align*}. The expression becomes:

\begin{align*}\frac{x^2+2x}{x+3}\left(\frac{x+1}{x+1}\right)+ \frac{x}{(x+3)(x+1)}\end{align*}

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

Now add the numerators and write as one rational expression:

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

Simplify the numerator by multiplying, combining like terms, and factoring if possible (the denominator is left in factored form):

\begin{align*}\frac{x^3+3x^2+3x}{(x+3)(x+1)}\end{align*}

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

The rational expression cannot be simplified any further so this is your answer. The restrictions are \begin{align*}x\ne -3\end{align*} and \begin{align*}x\ne -1\end{align*} because those values would cause one or both of the original denominators to be equal to zero.

Identify the lowest common denominator (LCD) in factored form.

\begin{align*}\frac{2x-3}{x^2-7x+10}-\frac{x-5}{x^2-2x-15}\end{align*}

To determine the LCD, begin by factoring the denominators.

 \begin{align*}\frac{2x-3}{x^2-7x+10}-\frac{x-5}{x^2-2x-15}=\frac{2x-3}{(x-5)(x-2)}-\frac{x-5}{(x-5)(x+3)}\end{align*}

The LCD is \begin{align*}\boxed{(x-5)(x-2)(x+3)}\end{align*}

\begin{align*}\frac{2x+1}{x^2+6x+9}+\frac{3x-2}{x^2+x-6}\end{align*}ii) 

 \begin{align*}\frac{2x+1}{x^2+6x+9}+\frac{3x-2}{x^2+x-6}=\frac{2x+1}{(x+3)(x+3)}+\frac{3x-2}{(x+3)(x-2)}\end{align*}

The LCD is \begin{align*}\boxed{(x+3)(x+3)(x-2)}\end{align*}

Add the following rational expressions and state the restrictions.

\begin{align*}\frac{3x+1}{x^2+8x+16}+\frac{2x-3}{x^2+x-12}\end{align*}

Begin by determining the LCD. Factor the denominators of each expression.

\begin{align*}\frac{3x+1}{x^2+8x+16}+\frac{2x-3}{x^2+x-12}=\frac{3x+1}{(x+4)(x+4)}+\frac{2x-3}{(x+4)(x-3)}\end{align*}

The LCD is \begin{align*}\boxed{(x+4)(x+4)(x-3)}\end{align*}

Multiply the numerators and denominators of each expression by the necessary factors to create the LCD.

\begin{align*}\frac{3x+1}{(x+4)(x+4)} {\color{red}\left(\frac{x-3}{x-3}\right)} + \frac{2x-3}{(x+4)(x-3)} {\color{red}\left(\frac{x+4}{x+4}\right)}\end{align*}

Multiply the numerators. Keep the denominators in factored form.

\begin{align*}\frac{3x^2-8x-3}{(x+4)(x+4)(x-3)}+\frac{2x^2+5x-12}{(x+4)(x+4)(x-3)}\end{align*}

Write the two expressions as one rational expression.

\begin{align*}\frac{3x^2-8x-3+2x^2+5x-12}{(x+4)(x+4)(x-3)}\end{align*}

Simplify the numerator by combining like terms.

\begin{align*}\frac{5x^2-3x-15}{(x+4)(x+4)(x-3)}\end{align*}

The numerator cannot be factored so the expression cannot be further simplified. The answer in lowest terms is:

\begin{align*}\boxed{\frac{5x^2-3x-15}{(x+4)(x+4)(x-3)}; x \ne -4; x \ne 3} \end{align*}

Subtract the following rational expressions and state the restrictions.

\begin{align*}\frac{x}{x^2-9x+18}-\frac{x-2}{x^2-10x+24}\end{align*}

Begin by determining the LCD. Factor the denominators of each expression.

\begin{align*}\frac{x}{(x-6)(x-3)}-\frac{x-2}{(x-6)(x-4)}\end{align*}

The LCD is \begin{align*}\boxed{(x-6)(x-3)(x-4)}\end{align*}

Multiply the numerators and denominators of each expression to get the LCD.

\begin{align*}\frac{x}{(x-6)(x-3)} {\color{red}\left(\frac{x-4}{x-4}\right)} - \frac{x-2}{(x-6)(x-4)} {\color{red}\left(\frac{x-3}{x-3}\right)}\end{align*}

Multiply the numerators.

\begin{align*}\frac{x^2-4x}{(x-6)(x-3)(x-4)} - \frac{x^2-5x+6}{(x-6)(x-3)(x-4)}\end{align*}

Write the expressions as one rational expression.

\begin{align*}\frac{x^2-4x-(x^2-5x+6)}{(x-6)(x-3)(x-4)}\end{align*}

\begin{align*}=\frac{x^2-4x-x^2+5x-6}{(x-6)(x-3)(x-4)}\end{align*}

Simplify the numerator by combining like terms.

\begin{align*}\frac{x-6}{(x-6)(x-3)(x-4)}\end{align*}

The term \begin{align*}(x-6)\end{align*} is common to both the numerator and the denominator. This term can be "cancelled." The solution is:

\begin{align*}\boxed{\frac{1}{(x-3)(x-4)};x \ne 3; x \ne 4; x \ne 6}\end{align*}

Examples

Example 1

Earlier, you were asked can you use your knowledge of rational expressions and adding fractions to add the following expressions. 

\begin{align*}\frac{3x}{x^2+6x-16}+\frac{2x}{x-2}\end{align*}

Factor the denominator of the first fraction and rewrite the problem:

\begin{align*}\frac{3x}{(x+8)(x-2)}+\frac{2x}{x-2}\end{align*}

The LCD is \begin{align*}(x+8)(x-2)\end{align*}.

\begin{align*}\frac{3x}{(x+8)(x-2)}+\frac{2x}{x-2} \left( {\color{red}\frac{x+8}{x+8}}\right)\end{align*}

Multiply the numerators.

\begin{align*}\frac{3x}{(x+8)(x-2)}+\frac{{\color{red}2x^2+16x}}{(x-2)(x+8)}\end{align*}

Write the two expressions a one rational expression.

\begin{align*}\frac{3x+2x^2+16x}{(x+8)(x-2)}\end{align*}

Simplify the numerator by combining like terms. Your final answer is:

\begin{align*}\boxed{\frac{2x^2+19x}{(x+8)(x-2)};x \ne -8; x \ne 2}\end{align*}

Example 2

Add or subtract the following and state the restrictions.

 \begin{align*}\frac{2x}{x^2-4}-\frac{1}{x-2}\end{align*}

\begin{align*}\frac{2x}{x^2-4}-\frac{1}{x-2}&=\frac{2x}{(x-2)(x+2)}-\frac{x+2}{(x-2)(x+2)}\\ &=\frac{2x-(x+2)}{(x-2)(x+2)}\\ &=\frac{(x-2)}{(x-2)(x+2)}\\ &=\frac{1}{(x+2)}; x \ne -2; x \ne 2\end{align*}

Example 3

Add or subtract the following and state the restrictions. 

 \begin{align*}\frac{-2}{3y^2+5y+2}+\frac{3}{y^2-7y-8}\end{align*}

\begin{align*}\frac{-2}{3y^2+5y+2}+\frac{3}{y^2-7y-8}&=\frac{-2}{(3y+2)(y+1)}+\frac{3}{(y-8)(y+1)}\\ &=\frac{-2(y-8)}{(3y+2)(y+1)(y-8)}+\frac{3(3y+2)}{(3y+2)(y-8)(y+1)}\\ &=\frac{-2y+16+9y+6}{(3y+2)(y-8)(y+1)}\\ &=\frac{7y+22}{(3y+2)(y-8)(y+1)}; y \ne -\frac{2}{3}; y \ne 8; y \ne -1\end{align*}

Example 4

\begin{align*}\frac{3m-1}{9m^3-36m^2} + \frac{2m+1}{2m^2-5m-12}\end{align*}

\begin{align*}\frac{3m-1}{9m^3-36m^2} + \frac{2m+1}{2m^2-5m-12}&=\frac{3m-1}{9m^2(m-4)} + \frac{2m+1}{(2m+3)(m-4)}\\ &=\frac{(3m-1)(2m+3)}{9m^2(m-4)(2m+3)} + \frac{9m^2(2m+1)}{9m^2(2m+3)(m-4)}\\ &=\frac{6m^2-2m+9m-3+18m^3+9m^2}{9m^2(m-4)(2m+3)}\\ &=\frac{18m^3+15m^2+7m-3}{9m^2(m-4)(2m+3)}; m \ne -\frac{3}{2}; m \ne 4; m \ne 0\end{align*}

Review

For each of the following rational expressions, determine the LCD.

  1. \begin{align*}\frac{2a-3}{4} + \frac{3a-1}{5} - \frac{a-5}{2}\end{align*}
  2. \begin{align*}\frac{5}{3x^2} - \frac{1}{2x} + \frac{3}{5x^3}\end{align*}
  3. \begin{align*}\frac{x}{a^2b} - \frac{y}{ab^2} + \frac{z}{3a^3b^2}\end{align*}
  4. \begin{align*}\frac{2w}{w^2-6w+5} - \frac{3w}{w^2-11w+30}\end{align*}
  5. \begin{align*}\frac{1}{y^2+5y} - \frac{2}{y^2+12y+35} - \frac{3}{y^3+7y^2}\end{align*}

For each of the following rational expressions, state the restrictions.

  1. \begin{align*}\frac{3}{x^2-5x+4} + \frac{4}{x^2-16}\end{align*}
  2. \begin{align*}\frac{5}{a^2+a} - \frac{2}{a^2+3a+2}\end{align*}
  3. \begin{align*}\frac{6}{m^2-5m} + \frac{7}{m^2-4m-5}\end{align*}
  4. \begin{align*}\frac{3n}{n^2+2n-3} - \frac{4n}{n^2+n-6}\end{align*}
  5. \begin{align*}\frac{6}{y^2-4} + \frac{4}{y^2+4y+4}\end{align*}

Add or subtract each of the following rational expressions and state the restrictions.

  1. \begin{align*}\frac{2a-3}{4} + \frac{3a-1}{5} - \frac{a-5}{2}\end{align*}
  2. \begin{align*}\frac{5}{3x^2} - \frac{1}{2x} + \frac{3}{5x^3}\end{align*}
  3. \begin{align*}\frac{x}{a^2b} - \frac{y}{ab^2} + \frac{z}{3a^3b^2}\end{align*}
  4. \begin{align*}\frac{2w}{w^2-6w+5} - \frac{3w}{w^2-11w+30}\end{align*}
  5. \begin{align*}\frac{1}{y^2+5y} - \frac{2}{y^2+12y+35} - \frac{3}{y^3+7y^2}\end{align*}

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 8.3. 

Vocabulary

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.

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