# Simplifying Rational Expressions

## Factor numerator and denominator and cancel

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Rational Expressions

### Rational Expression Simplification

Rational Number: A rational number is any number of the form \begin{align*}\frac{a}{b}\end{align*} , where \begin{align*}b \ne 0\end{align*} .

Rational Expression: A rational expression is any algebraic expression of the form \begin{align*}\frac{a(x)}{b(x)}\end{align*} , where \begin{align*}b \ne 0\end{align*} .

Tip: When simplifying rational expressions, it is important to use the fact that any number divided by itself is equal to 1.  This fact is our most effective tool in simplifying.

Tip: "Cancel out" when you can.  Note that you can only cancel out between multiplication, NOT addition or subtraction.

[(x+5) * 9]/[(x+5) *17]

(a+b)/(17-a) You cannot cancel a out here because it is addition and subtraction, not muliplication.

#### Guided Practice

Factor this expression:

\begin{align*}\frac{4x^2+20x+24}{2x^2+8x+8}\end{align*}

1) Factor out the numerator and denominator completely.

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

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

2) Cancel out where applicable.

\begin{align*}\frac{4\cancel{(x+2)}(x+3)}{2\cancel{(x+2)}(x+2)}\end{align*}

3) Simplify the whole numbers.

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

4) Add restrictions.  Need a refresher?  Look here.

\begin{align*}x\ne -2\end{align*}

### Rational Expression Multiplication and Division

Be sure to state restrictions and simplify after you have multiplied/divided.

Tip: Multiply and divide rational expressions the same way you would fractions. Multiply across for multiplication.  Flip the second term and multiply across for division.

Practice multiplying the problem below using the following steps:

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

1. Factor all terms that can be factored.
2. Multiply across to create a big rational expression. (Leave in factored form.)
3. Simplify.
4. State restrictions.

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