Precalculus Examples

$\frac{x}{(x + 2) (x - 4)}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place $A$ .

$\frac{A}{x + 2}$

Step 1.2

$\frac{A}{x + 2} + \frac{B}{x - 4}$

Step 1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(x + 2) (x - 4)$ .

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

Step 1.4

Cancel the common factor of $x + 2$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

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

Step 1.5

Cancel the common factor of $x - 4$ .

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Step 1.5.1

Cancel the common factor.

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

Step 1.5.2

Divide $x$ by $1$ .

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

Step 1.6

Simplify each term.

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Step 1.6.1

Cancel the common factor of $x + 2$ .

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Step 1.6.1.1

Cancel the common factor.

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

Step 1.6.1.2

Divide $(A) (x - 4)$ by $1$ .

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

Step 1.6.2

Apply the distributive property.

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

Step 1.6.3

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

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

Step 1.6.4

Cancel the common factor of $x - 4$ .

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Step 1.6.4.1

Cancel the common factor.

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

Step 1.6.4.2

Divide $(B) (x + 2)$ by $1$ .

$x = A x - 4 A + (B) (x + 2)$

Step 1.6.5

Apply the distributive property.

$x = A x - 4 A + B x + B \cdot 2$

Step 1.6.6

Move $2$ to the left of $B$ .

$x = A x - 4 A + B x + 2 B$

Step 1.7

Move $- 4 A$ .

$x = A x + B x - 4 A + 2 B$

Step 2

Create equations for the partial fraction variables and use them to set up a system of equations.

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Step 2.1

Create an equation for the partial fraction variables by equating the coefficients of $x$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = A + B$

Step 2.2

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $x$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = - 4 A + 2 B$

Step 2.3

Set up the system of equations to find the coefficients of the partial fractions.

$1 = A + B$

$0 = - 4 A + 2 B$

$1 = A + B$

$0 = - 4 A + 2 B$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $A$ in $1 = A + B$ .

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Step 3.1.1

Rewrite the equation as $A + B = 1$ .

$A + B = 1$

$0 = - 4 A + 2 B$

Step 3.1.2

Subtract $B$ from both sides of the equation.

$A = 1 - B$

$0 = - 4 A + 2 B$

$A = 1 - B$

$0 = - 4 A + 2 B$

Step 3.2

Replace all occurrences of $A$ with $1 - B$ in each equation.

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Step 3.2.1

Replace all occurrences of $A$ in $0 = - 4 A + 2 B$ with $1 - B$ .

$0 = - 4 (1 - B) + 2 B$

$A = 1 - B$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $- 4 (1 - B) + 2 B$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Apply the distributive property.

$0 = - 4 \cdot 1 - 4 (- B) + 2 B$

$A = 1 - B$

Step 3.2.2.1.1.2

Multiply $- 4$ by $1$ .

$0 = - 4 - 4 (- B) + 2 B$

$A = 1 - B$

Step 3.2.2.1.1.3

Multiply $- 1$ by $- 4$ .

$0 = - 4 + 4 B + 2 B$

$A = 1 - B$

$0 = - 4 + 4 B + 2 B$

$A = 1 - B$

Step 3.2.2.1.2

Add $4 B$ and $2 B$ .

$0 = - 4 + 6 B$

$A = 1 - B$

$0 = - 4 + 6 B$

$A = 1 - B$

$0 = - 4 + 6 B$

$A = 1 - B$

$0 = - 4 + 6 B$

$A = 1 - B$

Step 3.3

Solve for $B$ in $0 = - 4 + 6 B$ .

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Step 3.3.1

Rewrite the equation as $- 4 + 6 B = 0$ .

$- 4 + 6 B = 0$

$A = 1 - B$

Step 3.3.2

Add $4$ to both sides of the equation.

$6 B = 4$

$A = 1 - B$

Step 3.3.3

Divide each term in $6 B = 4$ by $6$ and simplify.

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Step 3.3.3.1

Divide each term in $6 B = 4$ by $6$ .

$\frac{6 B}{6} = \frac{4}{6}$

$A = 1 - B$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $6$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{6 B}{6} = \frac{4}{6}$

$A = 1 - B$

Step 3.3.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{4}{6}$

$A = 1 - B$

$B = \frac{4}{6}$

$A = 1 - B$

$B = \frac{4}{6}$

$A = 1 - B$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Cancel the common factor of $4$ and $6$ .

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Step 3.3.3.3.1.1

Factor $2$ out of $4$ .

$B = \frac{2 (2)}{6}$

$A = 1 - B$

Step 3.3.3.3.1.2

Cancel the common factors.

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Step 3.3.3.3.1.2.1

Factor $2$ out of $6$ .

$B = \frac{2 \cdot 2}{2 \cdot 3}$

$A = 1 - B$

Step 3.3.3.3.1.2.2

Cancel the common factor.

$B = \frac{2 \cdot 2}{2 \cdot 3}$

$A = 1 - B$

Step 3.3.3.3.1.2.3

Rewrite the expression.

$B = \frac{2}{3}$

$A = 1 - B$

$B = \frac{2}{3}$

$A = 1 - B$

$B = \frac{2}{3}$

$A = 1 - B$

$B = \frac{2}{3}$

$A = 1 - B$

$B = \frac{2}{3}$

$A = 1 - B$

$B = \frac{2}{3}$

$A = 1 - B$

Step 3.4

Replace all occurrences of $B$ with $\frac{2}{3}$ in each equation.

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Step 3.4.1

Replace all occurrences of $B$ in $A = 1 - B$ with $\frac{2}{3}$ .

$A = 1 - (\frac{2}{3})$

$B = \frac{2}{3}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $1 - (\frac{2}{3})$ .

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Step 3.4.2.1.1

Write $1$ as a fraction with a common denominator.

$A = \frac{3}{3} - \frac{2}{3}$

$B = \frac{2}{3}$

Step 3.4.2.1.2

Combine the numerators over the common denominator.

$A = \frac{3 - 2}{3}$

$B = \frac{2}{3}$

Step 3.4.2.1.3

Subtract $2$ from $3$ .

$A = \frac{1}{3}$

$B = \frac{2}{3}$

$A = \frac{1}{3}$

$B = \frac{2}{3}$

$A = \frac{1}{3}$

$B = \frac{2}{3}$

$A = \frac{1}{3}$

$B = \frac{2}{3}$

Step 3.5

List all of the solutions.

$A = \frac{1}{3}, B = \frac{2}{3}$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x + 2} + \frac{B}{x - 4}$ with the values found for $A$ and $B$ .

$\frac{\frac{1}{3}}{x + 2} + \frac{\frac{2}{3}}{x - 4}$