Calculus Examples

- \frac{6 y}{(y + 4) (y - 2)}

$- \frac{6 y}{(y + 4) (y - 2)}$

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

$A$ .

\frac{A}{y + 4}

$\frac{A}{y + 4}$

Step 1.2

B

$B$ .

\frac{A}{y + 4} + \frac{B}{y - 2}

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

Step 1.3

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

(y + 4) (y - 2)

$(y + 4) (y - 2)$ .

- \frac{6 y (y + 4) (y - 2)}{(y + 4) (y - 2)} = \frac{(A) (y + 4) (y - 2)}{y + 4} + \frac{(B) (y + 4) (y - 2)}{y - 2}

$- \frac{6 y (y + 4) (y - 2)}{(y + 4) (y - 2)} = \frac{(A) (y + 4) (y - 2)}{y + 4} + \frac{(B) (y + 4) (y - 2)}{y - 2}$

Step 1.4

Cancel the common factor of

y + 4

$y + 4$ .

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Step 1.4.1

Cancel the common factor.

- \frac{6 y (y + 4) (y - 2)}{(y + 4) (y - 2)} = \frac{(A) (y + 4) (y - 2)}{y + 4} + \frac{(B) (y + 4) (y - 2)}{y - 2}

$- \frac{6 y (y + 4) (y - 2)}{(y + 4) (y - 2)} = \frac{(A) (y + 4) (y - 2)}{y + 4} + \frac{(B) (y + 4) (y - 2)}{y - 2}$

Step 1.4.2

Rewrite the expression.

- \frac{6 y (y - 2)}{y - 2} = \frac{(A) (y + 4) (y - 2)}{y + 4} + \frac{(B) (y + 4) (y - 2)}{y - 2}

$- \frac{6 y (y - 2)}{y - 2} = \frac{(A) (y + 4) (y - 2)}{y + 4} + \frac{(B) (y + 4) (y - 2)}{y - 2}$

- \frac{6 y (y - 2)}{y - 2} = \frac{(A) (y + 4) (y - 2)}{y + 4} + \frac{(B) (y + 4) (y - 2)}{y - 2}

$- \frac{6 y (y - 2)}{y - 2} = \frac{(A) (y + 4) (y - 2)}{y + 4} + \frac{(B) (y + 4) (y - 2)}{y - 2}$

Step 1.5

Cancel the common factor of

y - 2

$y - 2$ .

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Step 1.5.1

Cancel the common factor.

- \frac{6 y (y - 2)}{y - 2} = \frac{(A) (y + 4) (y - 2)}{y + 4} + \frac{(B) (y + 4) (y - 2)}{y - 2}

$- \frac{6 y (y - 2)}{y - 2} = \frac{(A) (y + 4) (y - 2)}{y + 4} + \frac{(B) (y + 4) (y - 2)}{y - 2}$

Step 1.5.2

Divide

6 y

$6 y$ by

1

$1$ .

- (6 y) = \frac{(A) (y + 4) (y - 2)}{y + 4} + \frac{(B) (y + 4) (y - 2)}{y - 2}

$- (6 y) = \frac{(A) (y + 4) (y - 2)}{y + 4} + \frac{(B) (y + 4) (y - 2)}{y - 2}$

- (6 y) = \frac{(A) (y + 4) (y - 2)}{y + 4} + \frac{(B) (y + 4) (y - 2)}{y - 2}

$- (6 y) = \frac{(A) (y + 4) (y - 2)}{y + 4} + \frac{(B) (y + 4) (y - 2)}{y - 2}$

Step 1.6

Multiply

6

$6$ by

- 1

$- 1$ .

- 6 y = \frac{(A) (y + 4) (y - 2)}{y + 4} + \frac{(B) (y + 4) (y - 2)}{y - 2}

$- 6 y = \frac{(A) (y + 4) (y - 2)}{y + 4} + \frac{(B) (y + 4) (y - 2)}{y - 2}$

Step 1.7

Simplify each term.

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Step 1.7.1

Cancel the common factor of

y + 4

$y + 4$ .

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Step 1.7.1.1

Cancel the common factor.

- 6 y = \frac{A (y + 4) (y - 2)}{y + 4} + \frac{(B) (y + 4) (y - 2)}{y - 2}

$- 6 y = \frac{A (y + 4) (y - 2)}{y + 4} + \frac{(B) (y + 4) (y - 2)}{y - 2}$

Step 1.7.1.2

Divide

(A) (y - 2)

$(A) (y - 2)$ by

1

$1$ .

- 6 y = (A) (y - 2) + \frac{(B) (y + 4) (y - 2)}{y - 2}

$- 6 y = (A) (y - 2) + \frac{(B) (y + 4) (y - 2)}{y - 2}$

- 6 y = (A) (y - 2) + \frac{(B) (y + 4) (y - 2)}{y - 2}

$- 6 y = (A) (y - 2) + \frac{(B) (y + 4) (y - 2)}{y - 2}$

Step 1.7.2

Apply the distributive property.

- 6 y = A y + A \cdot - 2 + \frac{(B) (y + 4) (y - 2)}{y - 2}

$- 6 y = A y + A \cdot - 2 + \frac{(B) (y + 4) (y - 2)}{y - 2}$

Step 1.7.3

Move

- 2

$- 2$ to the left of

A

$A$ .

- 6 y = A y - 2 \cdot A + \frac{(B) (y + 4) (y - 2)}{y - 2}

$- 6 y = A y - 2 \cdot A + \frac{(B) (y + 4) (y - 2)}{y - 2}$

Step 1.7.4

Cancel the common factor of

y - 2

$y - 2$ .

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Step 1.7.4.1

Cancel the common factor.

- 6 y = A y - 2 A + \frac{(B) (y + 4) (y - 2)}{y - 2}

$- 6 y = A y - 2 A + \frac{(B) (y + 4) (y - 2)}{y - 2}$

Step 1.7.4.2

Divide

(B) (y + 4)

$(B) (y + 4)$ by

1

$1$ .

- 6 y = A y - 2 A + (B) (y + 4)

$- 6 y = A y - 2 A + (B) (y + 4)$

- 6 y = A y - 2 A + (B) (y + 4)

$- 6 y = A y - 2 A + (B) (y + 4)$

Step 1.7.5

Apply the distributive property.

- 6 y = A y - 2 A + B y + B \cdot 4

$- 6 y = A y - 2 A + B y + B \cdot 4$

Step 1.7.6

Move

4

$4$ to the left of

B

$B$ .

- 6 y = A y - 2 A + B y + 4 B

$- 6 y = A y - 2 A + B y + 4 B$

- 6 y = A y - 2 A + B y + 4 B

$- 6 y = A y - 2 A + B y + 4 B$

Step 1.8

Move

- 2 A

$- 2 A$ .

- 6 y = A y + B y - 2 A + 4 B

$- 6 y = A y + B y - 2 A + 4 B$

- 6 y = A y + B y - 2 A + 4 B

$- 6 y = A y + B y - 2 A + 4 B$

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

y

$y$ 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 = A + B

$- 6 = A + B$

Step 2.2

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

y

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

0 = - 2 A + 4 B

$0 = - 2 A + 4 B$

Step 2.3

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

- 6 = A + B

$- 6 = A + B$

0 = - 2 A + 4 B

$0 = - 2 A + 4 B$

- 6 = A + B

$- 6 = A + B$

0 = - 2 A + 4 B

$0 = - 2 A + 4 B$

Step 3

Solve the system of equations.

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Step 3.1

Solve for

A

$A$ in

- 6 = A + B

$- 6 = A + B$ .

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Step 3.1.1

Rewrite the equation as

A + B = - 6

$A + B = - 6$ .

A + B = - 6

$A + B = - 6$

0 = - 2 A + 4 B

$0 = - 2 A + 4 B$

Step 3.1.2

Subtract

B

$B$ from both sides of the equation.

A = - 6 - B

$A = - 6 - B$

0 = - 2 A + 4 B

$0 = - 2 A + 4 B$

A = - 6 - B

$A = - 6 - B$

0 = - 2 A + 4 B

$0 = - 2 A + 4 B$

Step 3.2

Replace all occurrences of

A

$A$ with

- 6 - B

$- 6 - B$ in each equation.

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Step 3.2.1

Replace all occurrences of

A

$A$ in

0 = - 2 A + 4 B

$0 = - 2 A + 4 B$ with

- 6 - B

$- 6 - B$ .

0 = - 2 (- 6 - B) + 4 B

$0 = - 2 (- 6 - B) + 4 B$

A = - 6 - B

$A = - 6 - B$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify

- 2 (- 6 - B) + 4 B

$- 2 (- 6 - B) + 4 B$ .

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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 = - 2 \cdot - 6 - 2 (- B) + 4 B

$0 = - 2 \cdot - 6 - 2 (- B) + 4 B$

A = - 6 - B

$A = - 6 - B$

Step 3.2.2.1.1.2

Multiply

- 2

$- 2$ by

- 6

$- 6$ .

0 = 12 - 2 (- B) + 4 B

$0 = 12 - 2 (- B) + 4 B$

A = - 6 - B

$A = - 6 - B$

Step 3.2.2.1.1.3

Multiply

- 1

$- 1$ by

- 2

$- 2$ .

0 = 12 + 2 B + 4 B

$0 = 12 + 2 B + 4 B$

A = - 6 - B

$A = - 6 - B$

0 = 12 + 2 B + 4 B

$0 = 12 + 2 B + 4 B$

A = - 6 - B

$A = - 6 - B$

Step 3.2.2.1.2

Add

2 B

$2 B$ and

4 B

$4 B$ .

0 = 12 + 6 B

$0 = 12 + 6 B$

A = - 6 - B

$A = - 6 - B$

0 = 12 + 6 B

$0 = 12 + 6 B$

A = - 6 - B

$A = - 6 - B$

0 = 12 + 6 B

$0 = 12 + 6 B$

A = - 6 - B

$A = - 6 - B$

0 = 12 + 6 B

$0 = 12 + 6 B$

A = - 6 - B

$A = - 6 - B$

Step 3.3

Solve for

B

$B$ in

0 = 12 + 6 B

$0 = 12 + 6 B$ .

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Step 3.3.1

Rewrite the equation as

12 + 6 B = 0

$12 + 6 B = 0$ .

12 + 6 B = 0

$12 + 6 B = 0$

A = - 6 - B

$A = - 6 - B$

Step 3.3.2

Subtract

12

$12$ from both sides of the equation.

6 B = - 12

$6 B = - 12$

A = - 6 - B

$A = - 6 - B$

Step 3.3.3

Divide each term in

6 B = - 12

$6 B = - 12$ by

6

$6$ and simplify.

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Step 3.3.3.1

Divide each term in

6 B = - 12

$6 B = - 12$ by

6

$6$ .

\frac{6 B}{6} = \frac{- 12}{6}

$\frac{6 B}{6} = \frac{- 12}{6}$

A = - 6 - B

$A = - 6 - B$

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

$6$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

\frac{6 B}{6} = \frac{- 12}{6}

$\frac{6 B}{6} = \frac{- 12}{6}$

A = - 6 - B

$A = - 6 - B$

Step 3.3.3.2.1.2

Divide

B

$B$ by

1

$1$ .

B = \frac{- 12}{6}

$B = \frac{- 12}{6}$

A = - 6 - B

$A = - 6 - B$

B = \frac{- 12}{6}

$B = \frac{- 12}{6}$

A = - 6 - B

$A = - 6 - B$

B = \frac{- 12}{6}

$B = \frac{- 12}{6}$

A = - 6 - B

$A = - 6 - B$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide

- 12

$- 12$ by

6

$6$ .

B = - 2

$B = - 2$

A = - 6 - B

$A = - 6 - B$

B = - 2

$B = - 2$

A = - 6 - B

$A = - 6 - B$

B = - 2

$B = - 2$

A = - 6 - B

$A = - 6 - B$

B = - 2

$B = - 2$

A = - 6 - B

$A = - 6 - B$

Step 3.4

Replace all occurrences of

B

$B$ with

- 2

$- 2$ in each equation.

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Step 3.4.1

Replace all occurrences of

B

$B$ in

A = - 6 - B

$A = - 6 - B$ with

- 2

$- 2$ .

A = - 6 - (- 2)

$A = - 6 - (- 2)$

B = - 2

$B = - 2$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify

- 6 - (- 2)

$- 6 - (- 2)$ .

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Step 3.4.2.1.1

Multiply

- 1

$- 1$ by

- 2

$- 2$ .

A = - 6 + 2

$A = - 6 + 2$

B = - 2

$B = - 2$

Step 3.4.2.1.2

Add

- 6

$- 6$ and

2

$2$ .

A = - 4

$A = - 4$

B = - 2

$B = - 2$

A = - 4

$A = - 4$

B = - 2

$B = - 2$

A = - 4

$A = - 4$

B = - 2

$B = - 2$

A = - 4

$A = - 4$

B = - 2

$B = - 2$

Step 3.5

List all of the solutions.

A = - 4, B = - 2

$A = - 4, B = - 2$

A = - 4, B = - 2

$A = - 4, B = - 2$

Step 4

Replace each of the partial fraction coefficients in

\frac{A}{y + 4} + \frac{B}{y - 2}

$\frac{A}{y + 4} + \frac{B}{y - 2}$ with the values found for

A

$A$ and

B

$B$ .

\frac{- 4}{y + 4} + \frac{- 2}{y - 2}

$\frac{- 4}{y + 4} + \frac{- 2}{y - 2}$

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