Precalculus Examples

\frac{4}{x (x - 1)}

$\frac{4}{x (x - 1)}$

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

B

$B$ .

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

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

Step 1.2

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is

x (x - 1)

$x (x - 1)$ .

\frac{4 (x (x - 1))}{x (x - 1)} = \frac{A (x (x - 1))}{x} + \frac{(B) (x (x - 1))}{x - 1}

$\frac{4 (x (x - 1))}{x (x - 1)} = \frac{A (x (x - 1))}{x} + \frac{(B) (x (x - 1))}{x - 1}$

Step 1.3

Cancel the common factor of

x

$x$ .

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Step 1.3.1

Cancel the common factor.

\frac{4 (x (x - 1))}{x (x - 1)} = \frac{A (x (x - 1))}{x} + \frac{(B) (x (x - 1))}{x - 1}

$\frac{4 (x (x - 1))}{x (x - 1)} = \frac{A (x (x - 1))}{x} + \frac{(B) (x (x - 1))}{x - 1}$

Step 1.3.2

Rewrite the expression.

\frac{4 (x - 1)}{x - 1} = \frac{A (x (x - 1))}{x} + \frac{(B) (x (x - 1))}{x - 1}

$\frac{4 (x - 1)}{x - 1} = \frac{A (x (x - 1))}{x} + \frac{(B) (x (x - 1))}{x - 1}$

\frac{4 (x - 1)}{x - 1} = \frac{A (x (x - 1))}{x} + \frac{(B) (x (x - 1))}{x - 1}

$\frac{4 (x - 1)}{x - 1} = \frac{A (x (x - 1))}{x} + \frac{(B) (x (x - 1))}{x - 1}$

Step 1.4

Cancel the common factor of

x - 1

$x - 1$ .

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Step 1.4.1

Cancel the common factor.

\frac{4 (x - 1)}{x - 1} = \frac{A (x (x - 1))}{x} + \frac{(B) (x (x - 1))}{x - 1}

$\frac{4 (x - 1)}{x - 1} = \frac{A (x (x - 1))}{x} + \frac{(B) (x (x - 1))}{x - 1}$

Step 1.4.2

Divide

4

$4$ by

1

$1$ .

4 = \frac{A (x (x - 1))}{x} + \frac{(B) (x (x - 1))}{x - 1}

$4 = \frac{A (x (x - 1))}{x} + \frac{(B) (x (x - 1))}{x - 1}$

4 = \frac{A (x (x - 1))}{x} + \frac{(B) (x (x - 1))}{x - 1}

$4 = \frac{A (x (x - 1))}{x} + \frac{(B) (x (x - 1))}{x - 1}$

Step 1.5

Simplify each term.

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Step 1.5.1

Cancel the common factor of

x

$x$ .

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Step 1.5.1.1

Cancel the common factor.

4 = \frac{A (x (x - 1))}{x} + \frac{(B) (x (x - 1))}{x - 1}

$4 = \frac{A (x (x - 1))}{x} + \frac{(B) (x (x - 1))}{x - 1}$

Step 1.5.1.2

Divide

A (x - 1)

$A (x - 1)$ by

1

$1$ .

4 = A (x - 1) + \frac{(B) (x (x - 1))}{x - 1}

$4 = A (x - 1) + \frac{(B) (x (x - 1))}{x - 1}$

4 = A (x - 1) + \frac{(B) (x (x - 1))}{x - 1}

$4 = A (x - 1) + \frac{(B) (x (x - 1))}{x - 1}$

Step 1.5.2

Apply the distributive property.

4 = A x + A \cdot - 1 + \frac{(B) (x (x - 1))}{x - 1}

$4 = A x + A \cdot - 1 + \frac{(B) (x (x - 1))}{x - 1}$

Step 1.5.3

Move

- 1

$- 1$ to the left of

A

$A$ .

4 = A x - 1 \cdot A + \frac{(B) (x (x - 1))}{x - 1}

$4 = A x - 1 \cdot A + \frac{(B) (x (x - 1))}{x - 1}$

Step 1.5.4

Rewrite

- 1 A

$- 1 A$ as

- A

$- A$ .

4 = A x - A + \frac{(B) (x (x - 1))}{x - 1}

$4 = A x - A + \frac{(B) (x (x - 1))}{x - 1}$

Step 1.5.5

Cancel the common factor of

x - 1

$x - 1$ .

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Step 1.5.5.1

Cancel the common factor.

4 = A x - A + \frac{B (x (x - 1))}{x - 1}

$4 = A x - A + \frac{B (x (x - 1))}{x - 1}$

Step 1.5.5.2

Divide

(B) (x)

$(B) (x)$ by

1

$1$ .

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

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

4 = A x - A + B x

$4 = A x - A + B x$

4 = A x - A + B x

$4 = A x - A + B x$

Step 1.6

Move

- A

$- A$ .

4 = A x + B x - A

$4 = A x + B x - A$

4 = A x + B x - A

$4 = A x + B x - A$

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

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

0 = A + B

$0 = A + B$

Step 2.2

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing

x

$x$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

4 = - 1 A

$4 = - 1 A$

Step 2.3

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

0 = A + B

$0 = A + B$

4 = - 1 A

$4 = - 1 A$

0 = A + B

$0 = A + B$

4 = - 1 A

$4 = - 1 A$

Step 3

Solve the system of equations.

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Step 3.1

Solve for

A

$A$ in

4 = - 1 A

$4 = - 1 A$ .

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Step 3.1.1

Rewrite the equation as

- 1 A = 4

$- 1 A = 4$ .

- 1 A = 4

$- 1 A = 4$

0 = A + B

$0 = A + B$

Step 3.1.2

Divide each term in

- 1 A = 4

$- 1 A = 4$ by

- 1

$- 1$ and simplify.

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Step 3.1.2.1

Divide each term in

- 1 A = 4

$- 1 A = 4$ by

- 1

$- 1$ .

\frac{- 1 A}{- 1} = \frac{4}{- 1}

$\frac{- 1 A}{- 1} = \frac{4}{- 1}$

0 = A + B

$0 = A + B$

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{4}{- 1}

$\frac{A}{1} = \frac{4}{- 1}$

0 = A + B

$0 = A + B$

Step 3.1.2.2.2

Divide

A

$A$ by

1

$1$ .

A = \frac{4}{- 1}

$A = \frac{4}{- 1}$

0 = A + B

$0 = A + B$

A = \frac{4}{- 1}

$A = \frac{4}{- 1}$

0 = A + B

$0 = A + B$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide

4

$4$ by

- 1

$- 1$ .

A = - 4

$A = - 4$

0 = A + B

$0 = A + B$

A = - 4

$A = - 4$

0 = A + B

$0 = A + B$

A = - 4

$A = - 4$

0 = A + B

$0 = A + B$

A = - 4

$A = - 4$

0 = A + B

$0 = A + B$

Step 3.2

Replace all occurrences of

A

$A$ with

- 4

$- 4$ in each equation.

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Step 3.2.1

Replace all occurrences of

A

$A$ in

0 = A + B

$0 = A + B$ with

- 4

$- 4$ .

0 = (- 4) + B

$0 = (- 4) + B$

A = - 4

$A = - 4$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Remove parentheses.

0 = - 4 + B

$0 = - 4 + B$

A = - 4

$A = - 4$

0 = - 4 + B

$0 = - 4 + B$

A = - 4

$A = - 4$

0 = - 4 + B

$0 = - 4 + B$

A = - 4

$A = - 4$

Step 3.3

Solve for

B

$B$ in

0 = - 4 + B

$0 = - 4 + B$ .

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Step 3.3.1

Rewrite the equation as

- 4 + B = 0

$- 4 + B = 0$ .

- 4 + B = 0

$- 4 + B = 0$

A = - 4

$A = - 4$

Step 3.3.2

Add

4

$4$ to both sides of the equation.

B = 4

$B = 4$

A = - 4

$A = - 4$

B = 4

$B = 4$

A = - 4

$A = - 4$

Step 3.4

Solve the system of equations.

$B = 4 A = - 4$

Step 3.5

List all of the solutions.

$B = 4, A = - 4$

Step 4

Replace each of the partial fraction coefficients in

$\frac{A}{x} + \frac{B}{x - 1}$ with the values found for

$A$ and

$B$ .

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