Precalculus Examples

$\frac{a x + b}{x^{2} - 16} = \frac{A}{x - 4} + \frac{B}{x + 4}$

Step 1

Multiply both sides by $x^{2} - 16$ .

$\frac{a x + b}{x^{2} - 16} (x^{2} - 16) = (\frac{A}{x - 4} + \frac{B}{x + 4}) (x^{2} - 16)$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $x^{2} - 16$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{a x + b}{x^{2} - 16} (x^{2} - 16) = (\frac{A}{x - 4} + \frac{B}{x + 4}) (x^{2} - 16)$

Step 2.1.1.2

Rewrite the expression.

$a x + b = (\frac{A}{x - 4} + \frac{B}{x + 4}) (x^{2} - 16)$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $(\frac{A}{x - 4} + \frac{B}{x + 4}) (x^{2} - 16)$ .

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Step 2.2.1.1

To write $\frac{A}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$a x + b = (\frac{A}{x - 4} \cdot \frac{x + 4}{x + 4} + \frac{B}{x + 4}) (x^{2} - 16)$

Step 2.2.1.2

To write $\frac{B}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$a x + b = (\frac{A}{x - 4} \cdot \frac{x + 4}{x + 4} + \frac{B}{x + 4} \cdot \frac{x - 4}{x - 4}) (x^{2} - 16)$

Step 2.2.1.3

Write each expression with a common denominator of $(x - 4) (x + 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.3.1

Multiply $\frac{A}{x - 4}$ by $\frac{x + 4}{x + 4}$ .

$a x + b = (\frac{A (x + 4)}{(x - 4) (x + 4)} + \frac{B}{x + 4} \cdot \frac{x - 4}{x - 4}) (x^{2} - 16)$

Step 2.2.1.3.2

Multiply $\frac{B}{x + 4}$ by $\frac{x - 4}{x - 4}$ .

$a x + b = (\frac{A (x + 4)}{(x - 4) (x + 4)} + \frac{B (x - 4)}{(x + 4) (x - 4)}) (x^{2} - 16)$

Step 2.2.1.3.3

Reorder the factors of $(x - 4) (x + 4)$ .

$a x + b = (\frac{A (x + 4)}{(x + 4) (x - 4)} + \frac{B (x - 4)}{(x + 4) (x - 4)}) (x^{2} - 16)$

Step 2.2.1.4

Combine the numerators over the common denominator.

$a x + b = \frac{A (x + 4) + B (x - 4)}{(x + 4) (x - 4)} (x^{2} - 16)$

Step 2.2.1.5

Simplify the numerator.

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Step 2.2.1.5.1

Apply the distributive property.

$a x + b = \frac{A x + A \cdot 4 + B (x - 4)}{(x + 4) (x - 4)} (x^{2} - 16)$

Step 2.2.1.5.2

Move $4$ to the left of $A$ .

$a x + b = \frac{A x + 4 \cdot A + B (x - 4)}{(x + 4) (x - 4)} (x^{2} - 16)$

Step 2.2.1.5.3

Apply the distributive property.

$a x + b = \frac{A x + 4 A + B x + B \cdot - 4}{(x + 4) (x - 4)} (x^{2} - 16)$

Step 2.2.1.5.4

Move $- 4$ to the left of $B$ .

$a x + b = \frac{A x + 4 A + B x - 4 B}{(x + 4) (x - 4)} (x^{2} - 16)$

Step 2.2.1.6

Multiply $\frac{A x + 4 A + B x - 4 B}{(x + 4) (x - 4)}$ by $x^{2} - 16$ .

$a x + b = \frac{(A x + 4 A + B x - 4 B) (x^{2} - 16)}{(x + 4) (x - 4)}$

Step 2.2.1.7

Simplify the numerator.

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Step 2.2.1.7.1

Rewrite $16$ as $4^{2}$ .

$a x + b = \frac{(A x + 4 A + B x - 4 B) (x^{2} - 4^{2})}{(x + 4) (x - 4)}$

Step 2.2.1.7.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

$a x + b = \frac{(A x + 4 A + B x - 4 B) (x + 4) (x - 4)}{(x + 4) (x - 4)}$

Step 2.2.1.8

Reduce the expression by cancelling the common factors.

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Step 2.2.1.8.1

Cancel the common factor of $x + 4$ .

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Step 2.2.1.8.1.1

Cancel the common factor.

$a x + b = \frac{(A x + 4 A + B x - 4 B) (x + 4) (x - 4)}{(x + 4) (x - 4)}$

Step 2.2.1.8.1.2

Rewrite the expression.

$a x + b = \frac{(A x + 4 A + B x - 4 B) (x - 4)}{x - 4}$

Step 2.2.1.8.2

Cancel the common factor of $x - 4$ .

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Step 2.2.1.8.2.1

Cancel the common factor.

$a x + b = \frac{(A x + 4 A + B x - 4 B) (x - 4)}{x - 4}$

Step 2.2.1.8.2.2

Divide $A x + 4 A + B x - 4 B$ by $1$ .

$a x + b = A x + 4 A + B x - 4 B$

Step 2.2.1.8.3

Simplify the expression.

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Step 2.2.1.8.3.1

Reorder $A$ and $x$ .

$a x + b = x A + 4 A + B x - 4 B$

Step 2.2.1.8.3.2

Reorder $B$ and $x$ .

$a x + b = x A + 4 A + x B - 4 B$

Step 2.2.1.8.3.3

Move $4 A$ .

$a x + b = x A + x B + 4 A - 4 B$

Step 3

Solve for $a$ .

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Step 3.1

Subtract $b$ from both sides of the equation.

$a x = x A + x B + 4 A - 4 B - b$

Step 3.2

Divide each term in $a x = x A + x B + 4 A - 4 B - b$ by $x$ and simplify.

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Step 3.2.1

Divide each term in $a x = x A + x B + 4 A - 4 B - b$ by $x$ .

$\frac{a x}{x} = \frac{x A}{x} + \frac{x B}{x} + \frac{4 A}{x} + \frac{- 4 B}{x} + \frac{- b}{x}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $x$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{a x}{x} = \frac{x A}{x} + \frac{x B}{x} + \frac{4 A}{x} + \frac{- 4 B}{x} + \frac{- b}{x}$

Step 3.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{x A}{x} + \frac{x B}{x} + \frac{4 A}{x} + \frac{- 4 B}{x} + \frac{- b}{x}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Cancel the common factor of $x$ .

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Step 3.2.3.1.1.1

Cancel the common factor.

$a = \frac{x A}{x} + \frac{x B}{x} + \frac{4 A}{x} + \frac{- 4 B}{x} + \frac{- b}{x}$

Step 3.2.3.1.1.2

Divide $A$ by $1$ .

$a = A + \frac{x B}{x} + \frac{4 A}{x} + \frac{- 4 B}{x} + \frac{- b}{x}$

Step 3.2.3.1.2

Cancel the common factor of $x$ .

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Step 3.2.3.1.2.1

Cancel the common factor.

$a = A + \frac{x B}{x} + \frac{4 A}{x} + \frac{- 4 B}{x} + \frac{- b}{x}$

Step 3.2.3.1.2.2

Divide $B$ by $1$ .

$a = A + B + \frac{4 A}{x} + \frac{- 4 B}{x} + \frac{- b}{x}$

Step 3.2.3.1.3

Move the negative in front of the fraction.

$a = A + B + \frac{4 A}{x} - \frac{4 B}{x} + \frac{- b}{x}$

Step 3.2.3.1.4

Move the negative in front of the fraction.

$a = A + B + \frac{4 A}{x} - \frac{4 B}{x} - \frac{b}{x}$