# Adding & Subtracting Rational Expressions

Rational expressions are fractions that have variables (including binomals or polynomials) in them.  The variables can be in the numerator or the denominator (or both!).

The rules for adding and subtracting rational expressions are exactly like those for adding and subtracting fractions.  If the denominators are the same, you can add or subtract the numerators.  If the denominators are different, you have to find a common denominator before you can add or subtract the numerators.

Let's review fraction addition and see how it compares to rational expression addition:

\require{cancel}\eqalign{\dfrac{4}{7}+\dfrac{2}{7}&=\dfrac{6}{7}}

Easy, right? The denominators are the same, so you add the numerators and use the same denominator.

The process is exactly the same with rational expressions, no matter how complicated the numerators and denominators are, if the denominators are the same, you can add the numerators and use the same denominator.

\eqalign{\dfrac{x^2+3x}{x^3+x^2+68}+\dfrac{z}{x^3+x^2+68}&=\dfrac{x^2+3x+z}{x^3+x^2+68}}

Essentially, if you remember how to add and subtract fractions, the process of adding and subtracting rational expressions is exactly the same.

Remember, when you add fractions you need to have a common denominator.

To get a common denominator, you find a denominator that both denominators go into evenly, then you convert the denominators.

You can multiply any number by 1.  And any fraction with the same number in the numerator and the denominator equals 1 (so, $\dfrac{3}{3} = 1$ and $\dfrac{68}{68}=1$, etc.).  So, if you figure out what you have to multiply your denomintor by to turn it into the common denominator that you've chosen (in this case, to turn 6 into 24 you need to muliply by 4 and to turn 8 into 24, you need to multiply by 3), and multiply each of your fractions by the fraction with those denominators that equals 1 (so, $\dfrac{4}{4}$ and $\dfrac{3}{3}$).

Then, you have two fractions, with the same denominator, that you can simply add together. Review the process with fractions first:

\eqalign{\dfrac{1}{6}+\dfrac{3}{8}\\ \text{Write it vertically }&\left\{ \begin{array}{rcl} \dfrac{1}{6}&\\ +\dfrac{3}{8}& \\\hline \qquad \qquad\\ \qquad \end{array}\right. \\\text{Find the common denominator: 24 }&\left\{ \begin{array}{rcl} \dfrac{1}{6}&=&&\dfrac{?}{24}\\+\dfrac{3}{8}&=&&\dfrac{?}{24}\\\hline \qquad \qquad \\ \qquad \end{array}\right.\\\begin{array}{rcl}\text{Multiply each fraction by a fractional version of 1 }\\\text{that will make each denominator equal 24 }\end{array}&\left\{ \begin{array}{rcl}\dfrac{1}{6}&\times&&\dfrac{4}{4}=&&&\dfrac{4}{24}\\+\dfrac{3}{8}&\times&&\dfrac{3}{3}=&&&\dfrac{9}{24}\\ \hline \qquad \qquad \end{array}\right.\\\text{Add the fractions with common denominators }&=\begin{array}&\quad && \quad &&& \quad &&&&\quad\dfrac{13}{24}\end {array}}

The process works the same exact way with rational expressions:

\eqalign{\dfrac{x}{6x}+\dfrac{x^2}{x-9}\\ \text{Write it vertically }&\left\{ \begin{array}{rcl}\dfrac{x}{6x}\;\;&\\ +\dfrac{x^2}{x-9}& \\\hline \qquad \qquad\\ \qquad \end{array}\right. \\\text{Find the common denominator: 6x(x-9) }&\left\{ \begin{array}{rcl}\dfrac{x}{6x}\;\;&=&&\dfrac{?}{6x(x-9)}\\+\dfrac{x^2}{(x-9)}&=&&\dfrac{?}{6x(x-9)}\\\hline \qquad \qquad \\ \qquad \end{array}\right.\\\begin{array}{rcl}\text{Multiply each fraction by a fraction = 1}\\\text{that will make each denominator equal 6x(x-9) }\end{array}&\left\{ \begin{array}{rcl}\dfrac{x}{6x}\;\;&\times&&\dfrac{x-9}{x-9}=&&&\dfrac{x(x-9)}{6x(x-9}\\+\dfrac{x^2}{x-9}&\times&&\;\;\dfrac{6x}{6x}=&&&\dfrac{x\times x^2}{6x(x-9)}\\ \hline \qquad \qquad \end{array}\right.\\\text{Add the fractions with common denominators }&=\begin{array}&\quad && \dfrac{x(x-9)+x^2\times 6x}{6x(x-9)}=\dfrac{6x^3+x^2-9x}{6x^2-54x}\end {array}}