Precalculus Examples

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

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

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{(4 - 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}

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

Step 1.4

Cancel the common factor of

y + 4

$y + 4$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

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

Step 1.5

Cancel the common factor of

$y + 2$ .

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Step 1.5.1

Cancel the common factor.

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

Step 1.5.2

Divide

$4 - y$ by

$1$ .

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

Step 1.6

Reorder

$4$ and

$- y$ .

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

Step 1.7

Simplify each term.

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Step 1.7.1

Cancel the common factor of

$y + 4$ .

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Step 1.7.1.1

Cancel the common factor.

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

Step 1.7.1.2

Divide

$(A) (y + 2)$ by

$1$ .

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

Step 1.7.2

Apply the distributive property.

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

Step 1.7.3

Move

$2$ to the left of

$A$ .

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

Step 1.7.4

Cancel the common factor of

$y + 2$ .

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Step 1.7.4.1

Cancel the common factor.

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

Step 1.7.4.2

Divide

$(B) (y + 4)$ by

$1$ .

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

Step 1.7.5

Apply the distributive property.

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

Step 1.7.6

Move

$4$ to the left of

$B$ .

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

Step 1.8

Move

$2 A$ .

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

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

$y$ 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

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

$4 = 2 A + 4 B$

Step 2.3

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

$- 1 = A + B$

$4 = 2 A + 4 B$

$- 1 = A + B$

$4 = 2 A + 4 B$

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

$A + B = - 1$ .

$A + B = - 1$

$4 = 2 A + 4 B$

Step 3.1.2

Subtract

$B$ from both sides of the equation.

$A = - 1 - B$

$4 = 2 A + 4 B$

$A = - 1 - B$

$4 = 2 A + 4 B$

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

$4 = 2 A + 4 B$ with

$- 1 - B$ .

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

$A = - 1 - B$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify

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

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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.

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

$A = - 1 - B$

Step 3.2.2.1.1.2

Multiply

$2$ by

$- 1$ .

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

$A = - 1 - B$

Step 3.2.2.1.1.3

Multiply

$- 1$ by

$2$ .

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

$A = - 1 - B$

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

$A = - 1 - B$

Step 3.2.2.1.2

Add

$- 2 B$ and

$4 B$ .

$4 = - 2 + 2 B$

$A = - 1 - B$

$4 = - 2 + 2 B$

$A = - 1 - B$

$4 = - 2 + 2 B$

$A = - 1 - B$

$4 = - 2 + 2 B$

$A = - 1 - B$

Step 3.3

Solve for

$B$ in

$4 = - 2 + 2 B$ .

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Step 3.3.1

Rewrite the equation as

$- 2 + 2 B = 4$ .

$- 2 + 2 B = 4$

$A = - 1 - B$

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

Add

$2$ to both sides of the equation.

$2 B = 4 + 2$

$A = - 1 - B$

Step 3.3.2.2

Add

$4$ and

$2$ .

$2 B = 6$

$A = - 1 - B$

$2 B = 6$

$A = - 1 - B$

Step 3.3.3

Divide each term in

$2 B = 6$ by

$2$ and simplify.

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Step 3.3.3.1

Divide each term in

$2 B = 6$ by

$2$ .

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

$A = - 1 - B$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of

$2$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

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

$A = - 1 - B$

Step 3.3.3.2.1.2

Divide

$B$ by

$1$ .

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

$A = - 1 - B$

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

$A = - 1 - B$

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

$A = - 1 - B$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide

$6$ by

$2$ .

$B = 3$

$A = - 1 - B$

$B = 3$

$A = - 1 - B$

$B = 3$

$A = - 1 - B$

$B = 3$

$A = - 1 - B$

Step 3.4

Replace all occurrences of

$B$ with

$3$ in each equation.

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Step 3.4.1

Replace all occurrences of

$B$ in

$A = - 1 - B$ with

$3$ .

$A = - 1 - (3)$

$B = 3$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify

$- 1 - (3)$ .

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Step 3.4.2.1.1

Multiply

$- 1$ by

$3$ .

$A = - 1 - 3$

$B = 3$

Step 3.4.2.1.2

Subtract

$3$ from

$- 1$ .

$A = - 4$

$B = 3$

$A = - 4$

$B = 3$

$A = - 4$

$B = 3$

$A = - 4$

$B = 3$

Step 3.5

List all of the solutions.

$A = - 4, B = 3$

Step 4

Replace each of the partial fraction coefficients in

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

$A$ and

$B$ .

$\frac{- 4}{y + 4} + \frac{3}{y + 2}$

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