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A rate of flow is modeled by $\frac{x ^{2} - 16}{x - 4}$ liters per second. What is the simplified expression?

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Answer:

Before you read

Answer first — see what you already know.

A concentration formula simplifies as $\frac{x ^{2} + 3 x - 10}{x - 2}$ . What is the simplified form?

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Answer:

Theory

What Is a Rational Expression?

A rational expression is a fraction where the numerator and/or denominator are polynomials:

$\frac{x^2 - 9}{x + 3} \qquad \frac{2x + 1}{x^2 - 4} \qquad \frac{3}{x - 5}$

The rules are the same as for numeric fractions — you can simplify by canceling common factors (not common terms!).

Simplifying a rational expression

Theory

Simplifying Rational Expressions

Steps:
1. Factor the numerator completely
2. Factor the denominator completely
3. Cancel any factors that appear in both

Critical rule: you can only cancel factors (things being multiplied), never terms (things being added/subtracted).

Example 1

Simplify $\frac{x^2 - 9}{x + 3}$

Factor numerator: $x^2 - 9 = (x+3)(x-3)$

$\frac{(x+3)(x-3)}{x+3}$

Cancel $(x+3)$ :

$= x - 3$

x - 3

(where

x \neq -3

)

Example 2

Simplify $\frac{x^2 + 5x + 6}{x^2 + 2x}$

Factor numerator: $x^2 + 5x + 6 = (x+2)(x+3)$

Factor denominator: $x^2 + 2x = x(x+2)$

$\frac{(x+2)(x+3)}{x(x+2)}$

Cancel $(x+2)$ :

$= \frac{x+3}{x}$

\frac{x+3}{x}

(where

x \neq 0, -2

)

3 more sections including examples, practice problems, and step-by-step solutions.

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