Precalculus Examples

$\frac{x^{2} - 6 x - 13}{(x + 2) (x^{2} - 1)}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

Factor the fraction.

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Step 1.1.1

Rewrite $1$ as $1^{2}$ .

$\frac{x^{2} - 6 x - 13}{(x + 2) (x^{2} - 1^{2})}$

Step 1.1.2

Factor.

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Step 1.1.2.1

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 $b = 1$ .

$\frac{x^{2} - 6 x - 13}{(x + 2) ((x + 1) (x - 1))}$

Step 1.1.2.2

Remove unnecessary parentheses.

$\frac{x^{2} - 6 x - 13}{(x + 2) (x + 1) (x - 1)}$

Step 1.2

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 1.3

$\frac{A}{x + 2} + \frac{B}{x + 1}$

Step 1.4

$\frac{A}{x + 2} + \frac{B}{x + 1} + \frac{C}{x - 1}$

Step 1.5

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(x + 2) (x + 1) (x - 1)$ .

$\frac{(x^{2} - 6 x - 13) (x + 2) (x + 1) (x - 1)}{(x + 2) (x + 1) (x - 1)} = \frac{(A) (x + 2) (x + 1) (x - 1)}{x + 2} + \frac{(B) (x + 2) (x + 1) (x - 1)}{x + 1} + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.6

Cancel the common factor of $x + 2$ .

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Step 1.6.1

Cancel the common factor.

Step 1.6.2

Rewrite the expression.

$\frac{((x^{2} - 6 x - 13) (x + 1)) (x - 1)}{(x + 1) (x - 1)} = \frac{(A) (x + 2) (x + 1) (x - 1)}{x + 2} + \frac{(B) (x + 2) (x + 1) (x - 1)}{x + 1} + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.7

Cancel the common factor of $x + 1$ .

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Step 1.7.1

Cancel the common factor.

$\frac{(x^{2} - 6 x - 13) (x + 1) (x - 1)}{(x + 1) (x - 1)} = \frac{(A) (x + 2) (x + 1) (x - 1)}{x + 2} + \frac{(B) (x + 2) (x + 1) (x - 1)}{x + 1} + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.7.2

Rewrite the expression.

$\frac{(x^{2} - 6 x - 13) (x - 1)}{x - 1} = \frac{(A) (x + 2) (x + 1) (x - 1)}{x + 2} + \frac{(B) (x + 2) (x + 1) (x - 1)}{x + 1} + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.8

Cancel the common factor of $x - 1$ .

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Step 1.8.1

Cancel the common factor.

$\frac{(x^{2} - 6 x - 13) (x - 1)}{x - 1} = \frac{(A) (x + 2) (x + 1) (x - 1)}{x + 2} + \frac{(B) (x + 2) (x + 1) (x - 1)}{x + 1} + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.8.2

Divide $x^{2} - 6 x - 13$ by $1$ .

$x^{2} - 6 x - 13 = \frac{(A) (x + 2) (x + 1) (x - 1)}{x + 2} + \frac{(B) (x + 2) (x + 1) (x - 1)}{x + 1} + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9

Simplify each term.

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Step 1.9.1

Cancel the common factor of $x + 2$ .

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Step 1.9.1.1

Cancel the common factor.

$x^{2} - 6 x - 13 = \frac{A (x + 2) (x + 1) (x - 1)}{x + 2} + \frac{(B) (x + 2) (x + 1) (x - 1)}{x + 1} + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.1.2

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

$x^{2} - 6 x - 13 = ((A) (x + 1)) (x - 1) + \frac{(B) (x + 2) (x + 1) (x - 1)}{x + 1} + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.2

Apply the distributive property.

$x^{2} - 6 x - 13 = (A x + A \cdot 1) (x - 1) + \frac{(B) (x + 2) (x + 1) (x - 1)}{x + 1} + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.3

Multiply $A$ by $1$ .

$x^{2} - 6 x - 13 = (A x + A) (x - 1) + \frac{(B) (x + 2) (x + 1) (x - 1)}{x + 1} + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.4

Expand $(A x + A) (x - 1)$ using the FOIL Method.

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Step 1.9.4.1

Apply the distributive property.

$x^{2} - 6 x - 13 = A x (x - 1) + A (x - 1) + \frac{(B) (x + 2) (x + 1) (x - 1)}{x + 1} + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.4.2

Apply the distributive property.

$x^{2} - 6 x - 13 = A x \cdot x + A x \cdot - 1 + A (x - 1) + \frac{(B) (x + 2) (x + 1) (x - 1)}{x + 1} + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.4.3

Apply the distributive property.

$x^{2} - 6 x - 13 = A x \cdot x + A x \cdot - 1 + A x + A \cdot - 1 + \frac{(B) (x + 2) (x + 1) (x - 1)}{x + 1} + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.5

Combine the opposite terms in $A x \cdot x + A x \cdot - 1 + A x + A \cdot - 1$ .

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Step 1.9.5.1

Reorder the factors in the terms $A x \cdot - 1$ and $A x$ .

$x^{2} - 6 x - 13 = A x \cdot x - A x + A x + A \cdot - 1 + \frac{(B) (x + 2) (x + 1) (x - 1)}{x + 1} + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.5.2

Add $- A x$ and $A x$ .

$x^{2} - 6 x - 13 = A x \cdot x + 0 + A \cdot - 1 + \frac{(B) (x + 2) (x + 1) (x - 1)}{x + 1} + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.5.3

Add $A x \cdot x$ and $0$ .

$x^{2} - 6 x - 13 = A x \cdot x + A \cdot - 1 + \frac{(B) (x + 2) (x + 1) (x - 1)}{x + 1} + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.6

Simplify each term.

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Step 1.9.6.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.9.6.1.1

Move $x$ .

$x^{2} - 6 x - 13 = A (x \cdot x) + A \cdot - 1 + \frac{(B) (x + 2) (x + 1) (x - 1)}{x + 1} + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.6.1.2

Multiply $x$ by $x$ .

$x^{2} - 6 x - 13 = A x^{2} + A \cdot - 1 + \frac{(B) (x + 2) (x + 1) (x - 1)}{x + 1} + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.6.2

Move $- 1$ to the left of $A$ .

$x^{2} - 6 x - 13 = A x^{2} - 1 \cdot A + \frac{(B) (x + 2) (x + 1) (x - 1)}{x + 1} + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.6.3

Rewrite $- 1 A$ as $- A$ .

$x^{2} - 6 x - 13 = A x^{2} - A + \frac{(B) (x + 2) (x + 1) (x - 1)}{x + 1} + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.7

Cancel the common factor of $x + 1$ .

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Step 1.9.7.1

Cancel the common factor.

$x^{2} - 6 x - 13 = A x^{2} - A + \frac{(B) (x + 2) (x + 1) (x - 1)}{x + 1} + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.7.2

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

$x^{2} - 6 x - 13 = A x^{2} - A + (B) (x + 2) (x - 1) + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.8

Apply the distributive property.

$x^{2} - 6 x - 13 = A x^{2} - A + (B x + B \cdot 2) (x - 1) + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.9

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

$x^{2} - 6 x - 13 = A x^{2} - A + (B x + 2 \cdot B) (x - 1) + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.10

Expand $(B x + 2 B) (x - 1)$ using the FOIL Method.

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Step 1.9.10.1

Apply the distributive property.

$x^{2} - 6 x - 13 = A x^{2} - A + B x (x - 1) + 2 B (x - 1) + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.10.2

Apply the distributive property.

$x^{2} - 6 x - 13 = A x^{2} - A + B x \cdot x + B x \cdot - 1 + 2 B (x - 1) + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.10.3

Apply the distributive property.

$x^{2} - 6 x - 13 = A x^{2} - A + B x \cdot x + B x \cdot - 1 + 2 B x + 2 B \cdot - 1 + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.11

Simplify and combine like terms.

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Step 1.9.11.1

Simplify each term.

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Step 1.9.11.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.9.11.1.1.1

Move $x$ .

$x^{2} - 6 x - 13 = A x^{2} - A + B (x \cdot x) + B x \cdot - 1 + 2 B x + 2 B \cdot - 1 + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.11.1.1.2

Multiply $x$ by $x$ .

$x^{2} - 6 x - 13 = A x^{2} - A + B x^{2} + B x \cdot - 1 + 2 B x + 2 B \cdot - 1 + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.11.1.2

Move $- 1$ to the left of $B x$ .

$x^{2} - 6 x - 13 = A x^{2} - A + B x^{2} - 1 \cdot (B x) + 2 B x + 2 B \cdot - 1 + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.11.1.3

Rewrite $- 1 (B x)$ as $- (B x)$ .

$x^{2} - 6 x - 13 = A x^{2} - A + B x^{2} - (B x) + 2 B x + 2 B \cdot - 1 + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.11.1.4

Multiply $- 1$ by $2$ .

$x^{2} - 6 x - 13 = A x^{2} - A + B x^{2} - B x + 2 B x - 2 B + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.11.2

Add $- B x$ and $2 B x$ .

$x^{2} - 6 x - 13 = A x^{2} - A + B x^{2} + 1 B x - 2 B + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.12

Multiply $B$ by $1$ .

$x^{2} - 6 x - 13 = A x^{2} - A + B x^{2} + B x - 2 B + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.13

Cancel the common factor of $x - 1$ .

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Step 1.9.13.1

Cancel the common factor.

$x^{2} - 6 x - 13 = A x^{2} - A + B x^{2} + B x - 2 B + \frac{(C) (x + 2) (x + 1) (x - 1)}{x - 1}$

Step 1.9.13.2

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

$x^{2} - 6 x - 13 = A x^{2} - A + B x^{2} + B x - 2 B + (C) (x + 2) (x + 1)$

Step 1.9.14

Apply the distributive property.

$x^{2} - 6 x - 13 = A x^{2} - A + B x^{2} + B x - 2 B + (C x + C \cdot 2) (x + 1)$

Step 1.9.15

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

$x^{2} - 6 x - 13 = A x^{2} - A + B x^{2} + B x - 2 B + (C x + 2 \cdot C) (x + 1)$

Step 1.9.16

Expand $(C x + 2 C) (x + 1)$ using the FOIL Method.

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Step 1.9.16.1

Apply the distributive property.

$x^{2} - 6 x - 13 = A x^{2} - A + B x^{2} + B x - 2 B + C x (x + 1) + 2 C (x + 1)$

Step 1.9.16.2

Apply the distributive property.

$x^{2} - 6 x - 13 = A x^{2} - A + B x^{2} + B x - 2 B + C x \cdot x + C x \cdot 1 + 2 C (x + 1)$

Step 1.9.16.3

Apply the distributive property.

$x^{2} - 6 x - 13 = A x^{2} - A + B x^{2} + B x - 2 B + C x \cdot x + C x \cdot 1 + 2 C x + 2 C \cdot 1$

Step 1.9.17

Simplify and combine like terms.

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Step 1.9.17.1

Simplify each term.

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Step 1.9.17.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.9.17.1.1.1

Move $x$ .

$x^{2} - 6 x - 13 = A x^{2} - A + B x^{2} + B x - 2 B + C (x \cdot x) + C x \cdot 1 + 2 C x + 2 C \cdot 1$

Step 1.9.17.1.1.2

Multiply $x$ by $x$ .

$x^{2} - 6 x - 13 = A x^{2} - A + B x^{2} + B x - 2 B + C x^{2} + C x \cdot 1 + 2 C x + 2 C \cdot 1$

Step 1.9.17.1.2

Multiply $C$ by $1$ .

$x^{2} - 6 x - 13 = A x^{2} - A + B x^{2} + B x - 2 B + C x^{2} + C x + 2 C x + 2 C \cdot 1$

Step 1.9.17.1.3

Multiply $2$ by $1$ .

$x^{2} - 6 x - 13 = A x^{2} - A + B x^{2} + B x - 2 B + C x^{2} + C x + 2 C x + 2 C$

Step 1.9.17.2

Add $C x$ and $2 C x$ .

$x^{2} - 6 x - 13 = A x^{2} - A + B x^{2} + B x - 2 B + C x^{2} + 3 C x + 2 C$

Step 1.10

Simplify the expression.

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Step 1.10.1

Reorder $B$ and $x$ .

$x^{2} - 6 x - 13 = A x^{2} - A + B x^{2} + B x - 2 B + C x^{2} + 3 C x + 2 C$

Step 1.10.2

Move $- 2 B$ .

$x^{2} - 6 x - 13 = A x^{2} - A + B x^{2} + B x + C x^{2} + 3 C x - 2 B + 2 C$

Step 1.10.3

Move $- A$ .

$x^{2} - 6 x - 13 = A x^{2} + B x^{2} + B x + C x^{2} + 3 C x - A - 2 B + 2 C$

Step 1.10.4

Move $B x$ .

$x^{2} - 6 x - 13 = A x^{2} + B x^{2} + C x^{2} + B x + 3 C x - A - 2 B + 2 C$

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 $x^{2}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = A + B + C$

Step 2.2

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.

$- 6 = B + 3 C$

Step 2.3

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.

$- 13 = - 1 A - 2 B + 2 C$

Step 2.4

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

$1 = A + B + C$

$- 6 = B + 3 C$

$- 13 = - 1 A - 2 B + 2 C$

$1 = A + B + C$

$- 6 = B + 3 C$

$- 13 = - 1 A - 2 B + 2 C$

Step 3

Solve the system of equations.

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Step 3.1

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

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Step 3.1.1

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

$A + B + C = 1$

$- 6 = B + 3 C$

$- 13 = - 1 A - 2 B + 2 C$

Step 3.1.2

Move all terms not containing $A$ to the right side of the equation.

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Step 3.1.2.1

Subtract $B$ from both sides of the equation.

$A + C = 1 - B$

$- 6 = B + 3 C$

$- 13 = - 1 A - 2 B + 2 C$

Step 3.1.2.2

Subtract $C$ from both sides of the equation.

$A = 1 - B - C$

$- 6 = B + 3 C$

$- 13 = - 1 A - 2 B + 2 C$

$A = 1 - B - C$

$- 6 = B + 3 C$

$- 13 = - 1 A - 2 B + 2 C$

$A = 1 - B - C$

$- 6 = B + 3 C$

$- 13 = - 1 A - 2 B + 2 C$

Step 3.2

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

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Step 3.2.1

Replace all occurrences of $A$ in $- 13 = - 1 A - 2 B + 2 C$ with $1 - B - C$ .

$- 13 = - 1 (1 - B - C) - 2 B + 2 C$

$A = 1 - B - C$

$- 6 = B + 3 C$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

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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.

$- 13 = - 1 \cdot 1 - 1 (- B) - 1 (- C) - 2 B + 2 C$

$A = 1 - B - C$

$- 6 = B + 3 C$

Step 3.2.2.1.1.2

Simplify.

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Step 3.2.2.1.1.2.1

Multiply $- 1$ by $1$ .

$- 13 = - 1 - 1 (- B) - 1 (- C) - 2 B + 2 C$

$A = 1 - B - C$

$- 6 = B + 3 C$

Step 3.2.2.1.1.2.2

Multiply $- 1 (- B)$ .

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Step 3.2.2.1.1.2.2.1

Multiply $- 1$ by $- 1$ .

$- 13 = - 1 + 1 B - 1 (- C) - 2 B + 2 C$

$A = 1 - B - C$

$- 6 = B + 3 C$

Step 3.2.2.1.1.2.2.2

Multiply $B$ by $1$ .

$- 13 = - 1 + B - 1 (- C) - 2 B + 2 C$

$A = 1 - B - C$

$- 6 = B + 3 C$

$- 13 = - 1 + B - 1 (- C) - 2 B + 2 C$

$A = 1 - B - C$

$- 6 = B + 3 C$

Step 3.2.2.1.1.2.3

Multiply $- 1 (- C)$ .

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Step 3.2.2.1.1.2.3.1

Multiply $- 1$ by $- 1$ .

$- 13 = - 1 + B + 1 C - 2 B + 2 C$

$A = 1 - B - C$

$- 6 = B + 3 C$

Step 3.2.2.1.1.2.3.2

Multiply $C$ by $1$ .

$- 13 = - 1 + B + C - 2 B + 2 C$

$A = 1 - B - C$

$- 6 = B + 3 C$

$- 13 = - 1 + B + C - 2 B + 2 C$

$A = 1 - B - C$

$- 6 = B + 3 C$

$- 13 = - 1 + B + C - 2 B + 2 C$

$A = 1 - B - C$

$- 6 = B + 3 C$

$- 13 = - 1 + B + C - 2 B + 2 C$

$A = 1 - B - C$

$- 6 = B + 3 C$

Step 3.2.2.1.2

Simplify by adding terms.

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Step 3.2.2.1.2.1

Subtract $2 B$ from $B$ .

$- 13 = - 1 - B + C + 2 C$

$A = 1 - B - C$

$- 6 = B + 3 C$

Step 3.2.2.1.2.2

Add $C$ and $2 C$ .

$- 13 = - 1 - B + 3 C$

$A = 1 - B - C$

$- 6 = B + 3 C$

$- 13 = - 1 - B + 3 C$

$A = 1 - B - C$

$- 6 = B + 3 C$

$- 13 = - 1 - B + 3 C$

$A = 1 - B - C$

$- 6 = B + 3 C$

$- 13 = - 1 - B + 3 C$

$A = 1 - B - C$

$- 6 = B + 3 C$

$- 13 = - 1 - B + 3 C$

$A = 1 - B - C$

$- 6 = B + 3 C$

Step 3.3

Solve for $B$ in $- 6 = B + 3 C$ .

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Step 3.3.1

Rewrite the equation as $B + 3 C = - 6$ .

$B + 3 C = - 6$

$- 13 = - 1 - B + 3 C$

$A = 1 - B - C$

Step 3.3.2

Subtract $3 C$ from both sides of the equation.

$B = - 6 - 3 C$

$- 13 = - 1 - B + 3 C$

$A = 1 - B - C$

$B = - 6 - 3 C$

$- 13 = - 1 - B + 3 C$

$A = 1 - B - C$

Step 3.4

Replace all occurrences of $B$ with $- 6 - 3 C$ in each equation.

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Step 3.4.1

Replace all occurrences of $B$ in $- 13 = - 1 - B + 3 C$ with $- 6 - 3 C$ .

$- 13 = - 1 - (- 6 - 3 C) + 3 C$

$B = - 6 - 3 C$

$A = 1 - B - C$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- 1 - (- 6 - 3 C) + 3 C$ .

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Step 3.4.2.1.1

Simplify each term.

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Step 3.4.2.1.1.1

Apply the distributive property.

$- 13 = - 1 + 6 - (- 3 C) + 3 C$

$B = - 6 - 3 C$

$A = 1 - B - C$

Step 3.4.2.1.1.2

Multiply $- 1$ by $- 6$ .

$- 13 = - 1 + 6 - (- 3 C) + 3 C$

$B = - 6 - 3 C$

$A = 1 - B - C$

Step 3.4.2.1.1.3

Multiply $- 3$ by $- 1$ .

$- 13 = - 1 + 6 + 3 C + 3 C$

$B = - 6 - 3 C$

$A = 1 - B - C$

$- 13 = - 1 + 6 + 3 C + 3 C$

$B = - 6 - 3 C$

$A = 1 - B - C$

Step 3.4.2.1.2

Simplify by adding terms.

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Step 3.4.2.1.2.1

Add $- 1$ and $6$ .

$- 13 = 5 + 3 C + 3 C$

$B = - 6 - 3 C$

$A = 1 - B - C$

Step 3.4.2.1.2.2

Add $3 C$ and $3 C$ .

$- 13 = 5 + 6 C$

$B = - 6 - 3 C$

$A = 1 - B - C$

$- 13 = 5 + 6 C$

$B = - 6 - 3 C$

$A = 1 - B - C$

$- 13 = 5 + 6 C$

$B = - 6 - 3 C$

$A = 1 - B - C$

$- 13 = 5 + 6 C$

$B = - 6 - 3 C$

$A = 1 - B - C$

Step 3.4.3

Replace all occurrences of $B$ in $A = 1 - B - C$ with $- 6 - 3 C$ .

$A = 1 - (- 6 - 3 C) - C$

$- 13 = 5 + 6 C$

$B = - 6 - 3 C$

Step 3.4.4

Simplify the right side.

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Step 3.4.4.1

Simplify $1 - (- 6 - 3 C) - C$ .

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Step 3.4.4.1.1

Simplify each term.

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Step 3.4.4.1.1.1

Apply the distributive property.

$A = 1 + 6 - (- 3 C) - C$

$- 13 = 5 + 6 C$

$B = - 6 - 3 C$

Step 3.4.4.1.1.2

Multiply $- 1$ by $- 6$ .

$A = 1 + 6 - (- 3 C) - C$

$- 13 = 5 + 6 C$

$B = - 6 - 3 C$

Step 3.4.4.1.1.3

Multiply $- 3$ by $- 1$ .

$A = 1 + 6 + 3 C - C$

$- 13 = 5 + 6 C$

$B = - 6 - 3 C$

$A = 1 + 6 + 3 C - C$

$- 13 = 5 + 6 C$

$B = - 6 - 3 C$

Step 3.4.4.1.2

Simplify by adding terms.

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Step 3.4.4.1.2.1

Add $1$ and $6$ .

$A = 7 + 3 C - C$

$- 13 = 5 + 6 C$

$B = - 6 - 3 C$

Step 3.4.4.1.2.2

Subtract $C$ from $3 C$ .

$A = 7 + 2 C$

$- 13 = 5 + 6 C$

$B = - 6 - 3 C$

$A = 7 + 2 C$

$- 13 = 5 + 6 C$

$B = - 6 - 3 C$

$A = 7 + 2 C$

$- 13 = 5 + 6 C$

$B = - 6 - 3 C$

$A = 7 + 2 C$

$- 13 = 5 + 6 C$

$B = - 6 - 3 C$

$A = 7 + 2 C$

$- 13 = 5 + 6 C$

$B = - 6 - 3 C$

Step 3.5

Solve for $C$ in $- 13 = 5 + 6 C$ .

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Step 3.5.1

Rewrite the equation as $5 + 6 C = - 13$ .

$5 + 6 C = - 13$

$A = 7 + 2 C$

$B = - 6 - 3 C$

Step 3.5.2

Move all terms not containing $C$ to the right side of the equation.

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Step 3.5.2.1

Subtract $5$ from both sides of the equation.

$6 C = - 13 - 5$

$A = 7 + 2 C$

$B = - 6 - 3 C$

Step 3.5.2.2

Subtract $5$ from $- 13$ .

$6 C = - 18$

$A = 7 + 2 C$

$B = - 6 - 3 C$

$6 C = - 18$

$A = 7 + 2 C$

$B = - 6 - 3 C$

Step 3.5.3

Divide each term in $6 C = - 18$ by $6$ and simplify.

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Step 3.5.3.1

Divide each term in $6 C = - 18$ by $6$ .

$\frac{6 C}{6} = \frac{- 18}{6}$

$A = 7 + 2 C$

$B = - 6 - 3 C$

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $6$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

$\frac{6 C}{6} = \frac{- 18}{6}$

$A = 7 + 2 C$

$B = - 6 - 3 C$

Step 3.5.3.2.1.2

Divide $C$ by $1$ .

$C = \frac{- 18}{6}$

$A = 7 + 2 C$

$B = - 6 - 3 C$

$C = \frac{- 18}{6}$

$A = 7 + 2 C$

$B = - 6 - 3 C$

$C = \frac{- 18}{6}$

$A = 7 + 2 C$

$B = - 6 - 3 C$

Step 3.5.3.3

Simplify the right side.

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Step 3.5.3.3.1

Divide $- 18$ by $6$ .

$C = - 3$

$A = 7 + 2 C$

$B = - 6 - 3 C$

$C = - 3$

$A = 7 + 2 C$

$B = - 6 - 3 C$

$C = - 3$

$A = 7 + 2 C$

$B = - 6 - 3 C$

$C = - 3$

$A = 7 + 2 C$

$B = - 6 - 3 C$

Step 3.6

Replace all occurrences of $C$ with $- 3$ in each equation.

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Step 3.6.1

Replace all occurrences of $C$ in $A = 7 + 2 C$ with $- 3$ .

$A = 7 + 2 (- 3)$

$C = - 3$

$B = - 6 - 3 C$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Simplify $7 + 2 (- 3)$ .

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Step 3.6.2.1.1

Multiply $2$ by $- 3$ .

$A = 7 - 6$

$C = - 3$

$B = - 6 - 3 C$

Step 3.6.2.1.2

Subtract $6$ from $7$ .

$A = 1$

$C = - 3$

$B = - 6 - 3 C$

$A = 1$

$C = - 3$

$B = - 6 - 3 C$

$A = 1$

$C = - 3$

$B = - 6 - 3 C$

Step 3.6.3

Replace all occurrences of $C$ in $B = - 6 - 3 C$ with $- 3$ .

$B = - 6 - 3 \cdot - 3$

$A = 1$

$C = - 3$

Step 3.6.4

Simplify the right side.

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Step 3.6.4.1

Simplify $- 6 - 3 \cdot - 3$ .

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Step 3.6.4.1.1

Multiply $- 3$ by $- 3$ .

$B = - 6 + 9$

$A = 1$

$C = - 3$

Step 3.6.4.1.2

Add $- 6$ and $9$ .

$B = 3$

$A = 1$

$C = - 3$

$B = 3$

$A = 1$

$C = - 3$

$B = 3$

$A = 1$

$C = - 3$

$B = 3$

$A = 1$

$C = - 3$

Step 3.7

List all of the solutions.

$B = 3, A = 1, C = - 3$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x + 2} + \frac{B}{x + 1} + \frac{C}{x - 1}$ with the values found for $A$ , $B$ , and $C$ .

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