## Simplifying Rational Expressions

A **rational expression** is a **fraction**. What makes a rational expression different from a simple or complex fraction is the **numerator and/or the denominator are polynomials**.

To simplify rational expressions, cancel out any **common factors**. Factors may be polynomials.

$\frac{x^{2} +5x +6}{x +2}=\frac{(x +3) (x +2)}{(x + 2)}=\frac{x+3}{1}=x + 3$

To simplify this rational expression, factor the trinomial in the numerator then cancel out the common factors.

## Adding and Subracting Rational Expressions

Find a common denominator for all fractions then add or subtract and simplify if needed.

There are many steps to solve adding and subtraction rational expression problems.

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

## Multiplying and Dividing Rational Expression

To make the computations less complicated, **cancel out common factors then multiply**.

For this multiplication problem, factor out the **common binomial terms** then multiply.

$\begin{align} \frac{x^{2} -x -2}{x-2}&\times\frac{x^{2} +4x +4}{x+2}=\\ \frac{(x+1)(x-2)}{x-2}&\times\frac{(x+2)(x+2)}{x+2}=\\ \frac{x+1}{1}&\times\frac{x+2}{1}=x^{2}+3x+2 \end{align}$

After canceling out any common factors, divide the rational expressions. Remember the **mnemonic device for dividing fractions**? **Keep it, switch it, flip it.**

$\begin{align} \frac{x^{2} +4x +4}{x+2}&\div\frac{x^{2} +6x +8}{x+2}=\\ \frac{(x+2)(x+2)}{x+2}&\div\frac{(x+2)(x+4)}{x+2}=\\ \frac{x+2}{1}&\div\frac{x+4}{1}=\\ \frac{x+2}{1}&\times\frac{1}{x+4}=\frac{x+2}{x+4} \end{align}$

## Longterm Division of Polynomials

**Long division of polynomials** follows similar steps as traditional long division. To check the solution, you can use the **Distributive Property**.

This quotient for this division problem is a **trinomial** and has no remainder.

$\begin{align} x^{2}+7x+12\\ x+6 ~\overline{\big)x^3+13x^{2}+54x +72}\\ \underline{-(x^3 +6x^{2})}\\ 7x^{2}+54x\\ \underline{-(7x^{2}+42x)}\\ 12x +72\\ \underline{-(12x+72)}\\ 0 \end{align}$

When when writing the dividend under the division bracket, include a placeholder for any powers that do not have terms. For this problem, notice the $0x^{3}$ term is a placeholder.

$\begin{align} 4x^{3}-8x^{2}+17x-32 +\frac{180}{(3x+6)}\\ 3x+6 ~\overline{\big) 12x^{4}+0x^3+3x^{2}+6x +12}\\ \underline{-(12x^4 +24x^{3})}\\ -24x^{3}+3x^{2}\\ \underline{-(-24x^{3}-48x^{2})}\\ 51x^{2} +6x\\ \underline{-(51x^{2}+102x)}\\ -96x -12\\ \underline{-(-96x -192)}\\ 180 \end{align}$

The quotient for this division problem did not work out evenly, and there is a **remainder**. The remainder is written as a fraction with the divisor as the denominator.