# Rational Expressions

A **rational expression** is an algebraic expression of the form $\frac{A}{B}$, where $A$ and $B$ are polynomials, and $B \neq 0$.

Here are a few examples of rational expressions where the denominator is simply $1$: $2x, 2x^2, 2x^2 +1.$ The following are a few examples of rational expressions where the denominator is a constant: $\frac{2x}{3}, \frac{2x^2}{5}, \frac{2x^2 +1}{4}.$ Also, the following are a few examples of rational expressions where the denominator contains variables: $\frac{1}{x}, \frac{x+1}{x}, \frac{x+1}{x-3}.$

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## Properties of Rational Expressions

Let $A$, $B$, and $C$ be real numbers or variable expressions, where $B \neq 0$ and $C \neq 0$.

$\frac{AC}{BC} = \frac{A}{B}$: You can divide out the top and bottom by a common factor $C$. This is also known as "canceling" $C$.

$\frac{A}{B} = \frac{A \times C}{B \times C}$: You can multiply the top and bottom by a common factor $C$.

## Simplifying Rational Expressions using the Laws of Exponents

## Simplify

$\frac{ 15xy^2 }{ 12y }.$

We have

$\frac{ 15xy^2 }{ 12y } = \frac{ 3 \cdot 5 xy^2 }{ 4 \cdot 3 y } = \frac{ 5 }{ 4 }xy^{2-1} = \frac{ 5 }{ 4 }xy. \ _\square$

## Simplify

$\left( \frac{a^5b^{-3}}{a^3b^8} \right)^2 .$

We have

$\left( \frac{a^5b^{-3}}{a^3b^8} \right)^2 = \left(a^{5-3}b^{-3-8}\right)^2 = \left(a^2b^{-11}\right)^2 = a^4b^{-22} = \frac{a^4}{b^{22}}. \ _\square$

For more examples applying the laws of exponents, see Simplifying Expressions with Exponents.

## Simplifying Rational Expressions by Factoring

## What is the value of

$\frac {x^2 - 9}{x + 3}$ at $x = 10$?

Factorizing the numerator of the expression gives $\frac {x^2 - 9}{x + 3} = \frac{(x-3)(x+3)}{x+3}.$ Canceling out the common factor $x + 3$, we get $\frac{(x-3)(x+3)}{x+3} = x - 3.$ For $x = 10$, we get $x - 3 = 10 - 3 = 7$. $_\square$

## Factor

$\frac{3x^3 - 6x^2}{9x^2}.$

Dividing both the numerator and denominator by a common factor of $3x^2$, we get

$\frac{3x^3 - 6x^2}{9x^2} = \frac{x - 2}{3}. \ _\square$

## Factor

$\frac{x^2 - x - 2}{x^2 - 2x}.$

The expression can be factored as

$\frac{x^2 - x - 2}{x^2 - 2x} =\frac{(x-2)(x+1)}{x(x-2)}.$

Canceling out the common factor $x - 2$,

$\frac{(x-2)(x+1)}{x(x-2)} = \frac{x+1}{x}.\ _\square$

## Factor

$\frac{6x^2 - x - 2}{10x^2 + 3x - 1} .$

We have

$\begin{aligned} \frac{6x^2 - x - 2}{10x^2 + 3x - 1} &=\frac{ (2x + 1)(3x - 2) }{ (2x + 1)(5x - 1) } \\ &= \frac{ 3x - 2 }{ 5x - 1 }.\ _\square \end{aligned}$

## Factor

$\frac{x^2 - y^2}{x^3 - y^3}.$

We have

$\frac{x^2 - y^2}{x^3 - y^3} =\frac{ (x - y)(x + y) }{ (x - y)(x^2 + xy + y^2) }.$

Canceling out the common factor $x - y$,

$\frac{ (x - y)(x + y) }{ (x - y)(x^2 + xy + y^2) } = \frac{x+y}{x^2 + xy + y^2}. \ _\square$

Next, see the Simplifying Rational Expressions page to learn how to multiply, divide, add, and subtract rational expressions.

**Cite as:**Rational Expressions.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/simplify-fractions/