# Rational Expressions

## 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.