Precalculus Examples

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

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

Factor $3$ out of $3 x^{2} - 6 x - 3$ .

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Step 1.1.1.1

Factor $3$ out of $3 x^{2}$ .

$\frac{3 (x^{2}) - 6 x - 3}{x^{3} - x}$

Step 1.1.1.2

Factor $3$ out of $- 6 x$ .

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

Step 1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (x^{2}) + 3 (- 2 x) + 3 (- 1)}{x^{3} - x}$

Step 1.1.1.4

Factor $3$ out of $3 (x^{2}) + 3 (- 2 x)$ .

$\frac{3 (x^{2} - 2 x) + 3 (- 1)}{x^{3} - x}$

Step 1.1.1.5

Factor $3$ out of $3 (x^{2} - 2 x) + 3 (- 1)$ .

$\frac{3 (x^{2} - 2 x - 1)}{x^{3} - x}$

Step 1.1.2

Factor $x$ out of $x^{3} - x$ .

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Step 1.1.2.1

Factor $x$ out of $x^{3}$ .

$\frac{3 (x^{2} - 2 x - 1)}{x \cdot x^{2} - x}$

Step 1.1.2.2

Factor $x$ out of $- x$ .

$\frac{3 (x^{2} - 2 x - 1)}{x \cdot x^{2} + x \cdot - 1}$

Step 1.1.2.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 1$ .

$\frac{3 (x^{2} - 2 x - 1)}{x (x^{2} - 1)}$

Step 1.1.3

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

$\frac{3 (x^{2} - 2 x - 1)}{x (x^{2} - 1^{2})}$

Step 1.1.4

Factor.

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Step 1.1.4.1

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 $b = 1$ .

$\frac{3 (x^{2} - 2 x - 1)}{x ((x + 1) (x - 1))}$

Step 1.1.4.2

Remove unnecessary parentheses.

$\frac{3 (x^{2} - 2 x - 1)}{x (x + 1) (x - 1)}$

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 $B$ .

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

Step 1.3

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

Step 1.4

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

$\frac{3 (x^{2} - 2 x - 1) (x (x + 1) (x - 1))}{x (x + 1) (x - 1)} = \frac{A (x (x + 1) (x - 1))}{x} + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.5

Cancel the common factor of $x$ .

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Step 1.5.1

Cancel the common factor.

$\frac{3 (x^{2} - 2 x - 1) (x (x + 1) (x - 1))}{x (x + 1) (x - 1)} = \frac{A (x (x + 1) (x - 1))}{x} + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.5.2

Rewrite the expression.

$\frac{3 (x^{2} - 2 x - 1) ((x + 1) (x - 1))}{(x + 1) (x - 1)} = \frac{A (x (x + 1) (x - 1))}{x} + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.6

Cancel the common factor of $x + 1$ .

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Step 1.6.1

Cancel the common factor.

$\frac{3 (x^{2} - 2 x - 1) ((x + 1) (x - 1))}{(x + 1) (x - 1)} = \frac{A (x (x + 1) (x - 1))}{x} + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.6.2

Rewrite the expression.

$\frac{3 (x^{2} - 2 x - 1) (x - 1)}{x - 1} = \frac{A (x (x + 1) (x - 1))}{x} + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.7

Cancel the common factor of $x - 1$ .

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Step 1.7.1

Cancel the common factor.

$\frac{3 (x^{2} - 2 x - 1) (x - 1)}{x - 1} = \frac{A (x (x + 1) (x - 1))}{x} + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.7.2

Divide $3 (x^{2} - 2 x - 1)$ by $1$ .

$3 (x^{2} - 2 x - 1) = \frac{A (x (x + 1) (x - 1))}{x} + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.8

Apply the distributive property.

$3 x^{2} + 3 (- 2 x) + 3 \cdot - 1 = \frac{A (x (x + 1) (x - 1))}{x} + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.9

Simplify.

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Step 1.9.1

Multiply $- 2$ by $3$ .

$3 x^{2} - 6 x + 3 \cdot - 1 = \frac{A (x (x + 1) (x - 1))}{x} + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.9.2

Multiply $3$ by $- 1$ .

$3 x^{2} - 6 x - 3 = \frac{A (x (x + 1) (x - 1))}{x} + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10

Simplify each term.

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Step 1.10.1

Cancel the common factor of $x$ .

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Step 1.10.1.1

Cancel the common factor.

$3 x^{2} - 6 x - 3 = \frac{A (x (x + 1) (x - 1))}{x} + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.1.2

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

$3 x^{2} - 6 x - 3 = A ((x + 1) (x - 1)) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.2

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

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Step 1.10.2.1

Apply the distributive property.

$3 x^{2} - 6 x - 3 = A (x (x - 1) + 1 (x - 1)) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.2.2

Apply the distributive property.

$3 x^{2} - 6 x - 3 = A (x \cdot x + x \cdot - 1 + 1 (x - 1)) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.2.3

Apply the distributive property.

$3 x^{2} - 6 x - 3 = A (x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.3

Simplify and combine like terms.

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Step 1.10.3.1

Simplify each term.

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Step 1.10.3.1.1

Multiply $x$ by $x$ .

$3 x^{2} - 6 x - 3 = A (x^{2} + x \cdot - 1 + 1 x + 1 \cdot - 1) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.3.1.2

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

$3 x^{2} - 6 x - 3 = A (x^{2} - 1 \cdot x + 1 x + 1 \cdot - 1) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.3.1.3

Rewrite $- 1 x$ as $- x$ .

$3 x^{2} - 6 x - 3 = A (x^{2} - x + 1 x + 1 \cdot - 1) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.3.1.4

Multiply $x$ by $1$ .

$3 x^{2} - 6 x - 3 = A (x^{2} - x + x + 1 \cdot - 1) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.3.1.5

Multiply $- 1$ by $1$ .

$3 x^{2} - 6 x - 3 = A (x^{2} - x + x - 1) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.3.2

Add $- x$ and $x$ .

$3 x^{2} - 6 x - 3 = A (x^{2} + 0 - 1) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.3.3

Add $x^{2}$ and $0$ .

$3 x^{2} - 6 x - 3 = A (x^{2} - 1) + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.4

Apply the distributive property.

$3 x^{2} - 6 x - 3 = A x^{2} + A \cdot - 1 + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.5

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

$3 x^{2} - 6 x - 3 = A x^{2} - 1 \cdot A + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.6

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

$3 x^{2} - 6 x - 3 = A x^{2} - A + \frac{(B) (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.7

Cancel the common factor of $x + 1$ .

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Step 1.10.7.1

Cancel the common factor.

$3 x^{2} - 6 x - 3 = A x^{2} - A + \frac{B (x (x + 1) (x - 1))}{x + 1} + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.7.2

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

$3 x^{2} - 6 x - 3 = A x^{2} - A + (B) (x (x - 1)) + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.8

Apply the distributive property.

$3 x^{2} - 6 x - 3 = A x^{2} - A + B (x \cdot x + x \cdot - 1) + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.9

Multiply $x$ by $x$ .

$3 x^{2} - 6 x - 3 = A x^{2} - A + B (x^{2} + x \cdot - 1) + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.10

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

$3 x^{2} - 6 x - 3 = A x^{2} - A + B (x^{2} - 1 \cdot x) + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.11

Rewrite $- 1 x$ as $- x$ .

$3 x^{2} - 6 x - 3 = A x^{2} - A + B (x^{2} - x) + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.12

Apply the distributive property.

$3 x^{2} - 6 x - 3 = A x^{2} - A + B x^{2} + B (- x) + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.13

Rewrite using the commutative property of multiplication.

$3 x^{2} - 6 x - 3 = A x^{2} - A + B x^{2} - B x + \frac{(C) (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.14

Cancel the common factor of $x - 1$ .

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Step 1.10.14.1

Cancel the common factor.

$3 x^{2} - 6 x - 3 = A x^{2} - A + B x^{2} - B x + \frac{C (x (x + 1) (x - 1))}{x - 1}$

Step 1.10.14.2

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

$3 x^{2} - 6 x - 3 = A x^{2} - A + B x^{2} - B x + (C) (x (x + 1))$

Step 1.10.15

Apply the distributive property.

$3 x^{2} - 6 x - 3 = A x^{2} - A + B x^{2} - B x + C (x \cdot x + x \cdot 1)$

Step 1.10.16

Multiply $x$ by $x$ .

$3 x^{2} - 6 x - 3 = A x^{2} - A + B x^{2} - B x + C (x^{2} + x \cdot 1)$

Step 1.10.17

Multiply $x$ by $1$ .

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

Step 1.10.18

Apply the distributive property.

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

Step 1.11

Simplify the expression.

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Step 1.11.1

Move $B$ .

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

Step 1.11.2

Move $- A$ .

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

Step 1.11.3

Move $- x B$ .

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

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.

$3 = A + B + C$

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.

$- 6 = - 1 B + C$

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.

$- 3 = - 1 A$

Step 2.4

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

$3 = A + B + C$

$- 6 = - 1 B + C$

$- 3 = - 1 A$

$3 = A + B + C$

$- 6 = - 1 B + C$

$- 3 = - 1 A$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $A$ in $- 3 = - 1 A$ .

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Step 3.1.1

Rewrite the equation as $- 1 A = - 3$ .

$- 1 A = - 3$

$3 = A + B + C$

$- 6 = - 1 B + C$

Step 3.1.2

Divide each term in $- 1 A = - 3$ by $- 1$ and simplify.

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Step 3.1.2.1

Divide each term in $- 1 A = - 3$ by $- 1$ .

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

$3 = A + B + C$

$- 6 = - 1 B + C$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Dividing two negative values results in a positive value.

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

$3 = A + B + C$

$- 6 = - 1 B + C$

Step 3.1.2.2.2

Divide $A$ by $1$ .

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

$3 = A + B + C$

$- 6 = - 1 B + C$

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

$3 = A + B + C$

$- 6 = - 1 B + C$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $- 3$ by $- 1$ .

$A = 3$

$3 = A + B + C$

$- 6 = - 1 B + C$

$A = 3$

$3 = A + B + C$

$- 6 = - 1 B + C$

$A = 3$

$3 = A + B + C$

$- 6 = - 1 B + C$

$A = 3$

$3 = A + B + C$

$- 6 = - 1 B + C$

Step 3.2

Replace all occurrences of $A$ with $3$ in each equation.

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Step 3.2.1

Replace all occurrences of $A$ in $3 = A + B + C$ with $3$ .

$3 = (3) + B + C$

$A = 3$

$- 6 = - 1 B + C$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Remove parentheses.

$3 = 3 + B + C$

$A = 3$

$- 6 = - 1 B + C$

$3 = 3 + B + C$

$A = 3$

$- 6 = - 1 B + C$

$3 = 3 + B + C$

$A = 3$

$- 6 = - 1 B + C$

Step 3.3

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

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Step 3.3.1

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

$3 + B + C = 3$

$A = 3$

$- 6 = - 1 B + C$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $3$ from both sides of the equation.

$B + C = 3 - 3$

$A = 3$

$- 6 = - 1 B + C$

Step 3.3.2.2

Subtract $C$ from both sides of the equation.

$B = 3 - 3 - C$

$A = 3$

$- 6 = - 1 B + C$

Step 3.3.2.3

Subtract $3$ from $3$ .

$B = 0 - C$

$A = 3$

$- 6 = - 1 B + C$

Step 3.3.2.4

Subtract $C$ from $0$ .

$B = - C$

$A = 3$

$- 6 = - 1 B + C$

$B = - C$

$A = 3$

$- 6 = - 1 B + C$

$B = - C$

$A = 3$

$- 6 = - 1 B + C$

Step 3.4

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

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Step 3.4.1

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

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

$B = - C$

$A = 3$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

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

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Step 3.4.2.1.1

Multiply $- 1 (- C)$ .

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Step 3.4.2.1.1.1

Multiply $- 1$ by $- 1$ .

$- 6 = 1 C + C$

$B = - C$

$A = 3$

Step 3.4.2.1.1.2

Multiply $C$ by $1$ .

$- 6 = C + C$

$B = - C$

$A = 3$

$- 6 = C + C$

$B = - C$

$A = 3$

Step 3.4.2.1.2

Add $C$ and $C$ .

$- 6 = 2 C$

$B = - C$

$A = 3$

$- 6 = 2 C$

$B = - C$

$A = 3$

$- 6 = 2 C$

$B = - C$

$A = 3$

$- 6 = 2 C$

$B = - C$

$A = 3$

Step 3.5

Solve for $C$ in $- 6 = 2 C$ .

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Step 3.5.1

Rewrite the equation as $2 C = - 6$ .

$2 C = - 6$

$B = - C$

$A = 3$

Step 3.5.2

Divide each term in $2 C = - 6$ by $2$ and simplify.

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Step 3.5.2.1

Divide each term in $2 C = - 6$ by $2$ .

$\frac{2 C}{2} = \frac{- 6}{2}$

$B = - C$

$A = 3$

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Cancel the common factor of $2$ .

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Step 3.5.2.2.1.1

Cancel the common factor.

$\frac{2 C}{2} = \frac{- 6}{2}$

$B = - C$

$A = 3$

Step 3.5.2.2.1.2

Divide $C$ by $1$ .

$C = \frac{- 6}{2}$

$B = - C$

$A = 3$

$C = \frac{- 6}{2}$

$B = - C$

$A = 3$

$C = \frac{- 6}{2}$

$B = - C$

$A = 3$

Step 3.5.2.3

Simplify the right side.

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Step 3.5.2.3.1

Divide $- 6$ by $2$ .

$C = - 3$

$B = - C$

$A = 3$

$C = - 3$

$B = - C$

$A = 3$

$C = - 3$

$B = - C$

$A = 3$

$C = - 3$

$B = - C$

$A = 3$

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 $B = - C$ with $- 3$ .

$B = - (- 3)$

$C = - 3$

$A = 3$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Multiply $- 1$ by $- 3$ .

$B = 3$

$C = - 3$

$A = 3$

$B = 3$

$C = - 3$

$A = 3$

$B = 3$

$C = - 3$

$A = 3$

Step 3.7

List all of the solutions.

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

Step 4

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

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