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: $$\frac{1}{x}, \frac{x+1}{x}, \frac{x+1}{x-3}.$$

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

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

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

Simplify

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

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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

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

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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.

What is the value of

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

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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

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

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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

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

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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

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

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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

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

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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.