Calculus Examples

$\frac{y}{(y - 2) (y + 3)}$

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 $A$ .

$\frac{A}{y - 2}$

Step 1.2

$\frac{A}{y - 2} + \frac{B}{y + 3}$

Step 1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(y - 2) (y + 3)$ .

$\frac{y (y - 2) (y + 3)}{(y - 2) (y + 3)} = \frac{(A) (y - 2) (y + 3)}{y - 2} + \frac{(B) (y - 2) (y + 3)}{y + 3}$

Step 1.4

Cancel the common factor of $y - 2$ .

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Step 1.4.1

Cancel the common factor.

$\frac{y (y - 2) (y + 3)}{(y - 2) (y + 3)} = \frac{(A) (y - 2) (y + 3)}{y - 2} + \frac{(B) (y - 2) (y + 3)}{y + 3}$

Step 1.4.2

Rewrite the expression.

$\frac{y (y + 3)}{y + 3} = \frac{(A) (y - 2) (y + 3)}{y - 2} + \frac{(B) (y - 2) (y + 3)}{y + 3}$

Step 1.5

Cancel the common factor of $y + 3$ .

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Step 1.5.1

Cancel the common factor.

$\frac{y (y + 3)}{y + 3} = \frac{(A) (y - 2) (y + 3)}{y - 2} + \frac{(B) (y - 2) (y + 3)}{y + 3}$

Step 1.5.2

Divide $y$ by $1$ .

$y = \frac{(A) (y - 2) (y + 3)}{y - 2} + \frac{(B) (y - 2) (y + 3)}{y + 3}$

Step 1.6

Simplify each term.

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Step 1.6.1

Cancel the common factor of $y - 2$ .

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Step 1.6.1.1

Cancel the common factor.

$y = \frac{A (y - 2) (y + 3)}{y - 2} + \frac{(B) (y - 2) (y + 3)}{y + 3}$

Step 1.6.1.2

Divide $(A) (y + 3)$ by $1$ .

$y = (A) (y + 3) + \frac{(B) (y - 2) (y + 3)}{y + 3}$

Step 1.6.2

Apply the distributive property.

$y = A y + A \cdot 3 + \frac{(B) (y - 2) (y + 3)}{y + 3}$

Step 1.6.3

Move $3$ to the left of $A$ .

$y = A y + 3 \cdot A + \frac{(B) (y - 2) (y + 3)}{y + 3}$

Step 1.6.4

Cancel the common factor of $y + 3$ .

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Step 1.6.4.1

Cancel the common factor.

$y = A y + 3 A + \frac{(B) (y - 2) (y + 3)}{y + 3}$

Step 1.6.4.2

Divide $(B) (y - 2)$ by $1$ .

$y = A y + 3 A + (B) (y - 2)$

Step 1.6.5

Apply the distributive property.

$y = A y + 3 A + B y + B \cdot - 2$

Step 1.6.6

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

$y = A y + 3 A + B y - 2 B$

Step 1.7

Move $3 A$ .

$y = A y + B y + 3 A - 2 B$

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.

$0 = 3 A - 2 B$

Step 2.3

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

$1 = A + B$

$0 = 3 A - 2 B$

$1 = A + B$

$0 = 3 A - 2 B$

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 $A + B = 1$ .

$A + B = 1$

$0 = 3 A - 2 B$

Step 3.1.2

Subtract $B$ from both sides of the equation.

$A = 1 - B$

$0 = 3 A - 2 B$

$A = 1 - B$

$0 = 3 A - 2 B$

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 $0 = 3 A - 2 B$ with $1 - B$ .

$0 = 3 (1 - B) - 2 B$

$A = 1 - B$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $3 (1 - B) - 2 B$ .

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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.

$0 = 3 \cdot 1 + 3 (- B) - 2 B$

$A = 1 - B$

Step 3.2.2.1.1.2

Multiply $3$ by $1$ .

$0 = 3 + 3 (- B) - 2 B$

$A = 1 - B$

Step 3.2.2.1.1.3

Multiply $- 1$ by $3$ .

$0 = 3 - 3 B - 2 B$

$A = 1 - B$

$0 = 3 - 3 B - 2 B$

$A = 1 - B$

Step 3.2.2.1.2

Subtract $2 B$ from $- 3 B$ .

$0 = 3 - 5 B$

$A = 1 - B$

$0 = 3 - 5 B$

$A = 1 - B$

$0 = 3 - 5 B$

$A = 1 - B$

$0 = 3 - 5 B$

$A = 1 - B$

Step 3.3

Solve for $B$ in $0 = 3 - 5 B$ .

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Step 3.3.1

Rewrite the equation as $3 - 5 B = 0$ .

$3 - 5 B = 0$

$A = 1 - B$

Step 3.3.2

Subtract $3$ from both sides of the equation.

$- 5 B = - 3$

$A = 1 - B$

Step 3.3.3

Divide each term in $- 5 B = - 3$ by $- 5$ and simplify.

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Step 3.3.3.1

Divide each term in $- 5 B = - 3$ by $- 5$ .

$\frac{- 5 B}{- 5} = \frac{- 3}{- 5}$

$A = 1 - B$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 5$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 5 B}{- 5} = \frac{- 3}{- 5}$

$A = 1 - B$

Step 3.3.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{- 3}{- 5}$

$A = 1 - B$

$B = \frac{- 3}{- 5}$

$A = 1 - B$

$B = \frac{- 3}{- 5}$

$A = 1 - B$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Dividing two negative values results in a positive value.

$B = \frac{3}{5}$

$A = 1 - B$

$B = \frac{3}{5}$

$A = 1 - B$

$B = \frac{3}{5}$

$A = 1 - B$

$B = \frac{3}{5}$

$A = 1 - B$

Step 3.4

Replace all occurrences of $B$ with $\frac{3}{5}$ in each equation.

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Step 3.4.1

Replace all occurrences of $B$ in $A = 1 - B$ with $\frac{3}{5}$ .

$A = 1 - (\frac{3}{5})$

$B = \frac{3}{5}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $1 - (\frac{3}{5})$ .

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Step 3.4.2.1.1

Write $1$ as a fraction with a common denominator.

$A = \frac{5}{5} - \frac{3}{5}$

$B = \frac{3}{5}$

Step 3.4.2.1.2

Combine the numerators over the common denominator.

$A = \frac{5 - 3}{5}$

$B = \frac{3}{5}$

Step 3.4.2.1.3

Subtract $3$ from $5$ .

$A = \frac{2}{5}$

$B = \frac{3}{5}$

$A = \frac{2}{5}$

$B = \frac{3}{5}$

$A = \frac{2}{5}$

$B = \frac{3}{5}$

$A = \frac{2}{5}$

$B = \frac{3}{5}$

Step 3.5

List all of the solutions.

$A = \frac{2}{5}, B = \frac{3}{5}$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{y - 2} + \frac{B}{y + 3}$ with the values found for $A$ and $B$ .

$\frac{\frac{2}{5}}{y - 2} + \frac{\frac{3}{5}}{y + 3}$

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