Precalculus Examples

$\frac{11 x - 10}{(x - 3) (x + 7)}$

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 $A$ .

$\frac{A}{x - 3}$

Step 1.2

$\frac{A}{x - 3} + \frac{B}{x + 7}$

Step 1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(x - 3) (x + 7)$ .

$\frac{(11 x - 10) (x - 3) (x + 7)}{(x - 3) (x + 7)} = \frac{(A) (x - 3) (x + 7)}{x - 3} + \frac{(B) (x - 3) (x + 7)}{x + 7}$

Step 1.4

Cancel the common factor of $x - 3$ .

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Step 1.4.1

Cancel the common factor.

$\frac{(11 x - 10) (x - 3) (x + 7)}{(x - 3) (x + 7)} = \frac{(A) (x - 3) (x + 7)}{x - 3} + \frac{(B) (x - 3) (x + 7)}{x + 7}$

Step 1.4.2

Rewrite the expression.

$\frac{(11 x - 10) (x + 7)}{x + 7} = \frac{(A) (x - 3) (x + 7)}{x - 3} + \frac{(B) (x - 3) (x + 7)}{x + 7}$

Step 1.5

Cancel the common factor of $x + 7$ .

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Step 1.5.1

Cancel the common factor.

$\frac{(11 x - 10) (x + 7)}{x + 7} = \frac{(A) (x - 3) (x + 7)}{x - 3} + \frac{(B) (x - 3) (x + 7)}{x + 7}$

Step 1.5.2

Divide $11 x - 10$ by $1$ .

$11 x - 10 = \frac{(A) (x - 3) (x + 7)}{x - 3} + \frac{(B) (x - 3) (x + 7)}{x + 7}$

Step 1.6

Simplify each term.

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Step 1.6.1

Cancel the common factor of $x - 3$ .

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Step 1.6.1.1

Cancel the common factor.

$11 x - 10 = \frac{A (x - 3) (x + 7)}{x - 3} + \frac{(B) (x - 3) (x + 7)}{x + 7}$

Step 1.6.1.2

Divide $(A) (x + 7)$ by $1$ .

$11 x - 10 = (A) (x + 7) + \frac{(B) (x - 3) (x + 7)}{x + 7}$

Step 1.6.2

Apply the distributive property.

$11 x - 10 = A x + A \cdot 7 + \frac{(B) (x - 3) (x + 7)}{x + 7}$

Step 1.6.3

Move $7$ to the left of $A$ .

$11 x - 10 = A x + 7 \cdot A + \frac{(B) (x - 3) (x + 7)}{x + 7}$

Step 1.6.4

Cancel the common factor of $x + 7$ .

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Step 1.6.4.1

Cancel the common factor.

$11 x - 10 = A x + 7 A + \frac{(B) (x - 3) (x + 7)}{x + 7}$

Step 1.6.4.2

Divide $(B) (x - 3)$ by $1$ .

$11 x - 10 = A x + 7 A + (B) (x - 3)$

Step 1.6.5

Apply the distributive property.

$11 x - 10 = A x + 7 A + B x + B \cdot - 3$

Step 1.6.6

Move $- 3$ to the left of $B$ .

$11 x - 10 = A x + 7 A + B x - 3 B$

Step 1.7

Move $7 A$ .

$11 x - 10 = A x + B x + 7 A - 3 B$

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

$11 = A + B$

Step 2.2

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.

$- 10 = 7 A - 3 B$

Step 2.3

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

$11 = A + B$

$- 10 = 7 A - 3 B$

$11 = A + B$

$- 10 = 7 A - 3 B$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $A$ in $11 = A + B$ .

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Step 3.1.1

Rewrite the equation as $A + B = 11$ .

$A + B = 11$

$- 10 = 7 A - 3 B$

Step 3.1.2

Subtract $B$ from both sides of the equation.

$A = 11 - B$

$- 10 = 7 A - 3 B$

$A = 11 - B$

$- 10 = 7 A - 3 B$

Step 3.2

Replace all occurrences of $A$ with $11 - B$ in each equation.

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Step 3.2.1

Replace all occurrences of $A$ in $- 10 = 7 A - 3 B$ with $11 - B$ .

$- 10 = 7 (11 - B) - 3 B$

$A = 11 - B$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $7 (11 - B) - 3 B$ .

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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.

$- 10 = 7 \cdot 11 + 7 (- B) - 3 B$

$A = 11 - B$

Step 3.2.2.1.1.2

Multiply $7$ by $11$ .

$- 10 = 77 + 7 (- B) - 3 B$

$A = 11 - B$

Step 3.2.2.1.1.3

Multiply $- 1$ by $7$ .

$- 10 = 77 - 7 B - 3 B$

$A = 11 - B$

$- 10 = 77 - 7 B - 3 B$

$A = 11 - B$

Step 3.2.2.1.2

Subtract $3 B$ from $- 7 B$ .

$- 10 = 77 - 10 B$

$A = 11 - B$

$- 10 = 77 - 10 B$

$A = 11 - B$

$- 10 = 77 - 10 B$

$A = 11 - B$

$- 10 = 77 - 10 B$

$A = 11 - B$

Step 3.3

Solve for $B$ in $- 10 = 77 - 10 B$ .

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Step 3.3.1

Rewrite the equation as $77 - 10 B = - 10$ .

$77 - 10 B = - 10$

$A = 11 - B$

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

Subtract $77$ from both sides of the equation.

$- 10 B = - 10 - 77$

$A = 11 - B$

Step 3.3.2.2

Subtract $77$ from $- 10$ .

$- 10 B = - 87$

$A = 11 - B$

$- 10 B = - 87$

$A = 11 - B$

Step 3.3.3

Divide each term in $- 10 B = - 87$ by $- 10$ and simplify.

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Step 3.3.3.1

Divide each term in $- 10 B = - 87$ by $- 10$ .

$\frac{- 10 B}{- 10} = \frac{- 87}{- 10}$

$A = 11 - B$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 10$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 10 B}{- 10} = \frac{- 87}{- 10}$

$A = 11 - B$

Step 3.3.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{- 87}{- 10}$

$A = 11 - B$

$B = \frac{- 87}{- 10}$

$A = 11 - B$

$B = \frac{- 87}{- 10}$

$A = 11 - B$

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.

$B = \frac{87}{10}$

$A = 11 - B$

$B = \frac{87}{10}$

$A = 11 - B$

$B = \frac{87}{10}$

$A = 11 - B$

$B = \frac{87}{10}$

$A = 11 - B$

Step 3.4

Replace all occurrences of $B$ with $\frac{87}{10}$ in each equation.

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Step 3.4.1

Replace all occurrences of $B$ in $A = 11 - B$ with $\frac{87}{10}$ .

$A = 11 - (\frac{87}{10})$

$B = \frac{87}{10}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $11 - (\frac{87}{10})$ .

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Step 3.4.2.1.1

To write $11$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$A = 11 \cdot \frac{10}{10} - \frac{87}{10}$

$B = \frac{87}{10}$

Step 3.4.2.1.2

Combine $11$ and $\frac{10}{10}$ .

$A = \frac{11 \cdot 10}{10} - \frac{87}{10}$

$B = \frac{87}{10}$

Step 3.4.2.1.3

Combine the numerators over the common denominator.

$A = \frac{11 \cdot 10 - 87}{10}$

$B = \frac{87}{10}$

Step 3.4.2.1.4

Simplify the numerator.

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Step 3.4.2.1.4.1

Multiply $11$ by $10$ .

$A = \frac{110 - 87}{10}$

$B = \frac{87}{10}$

Step 3.4.2.1.4.2

Subtract $87$ from $110$ .

$A = \frac{23}{10}$

$B = \frac{87}{10}$

$A = \frac{23}{10}$

$B = \frac{87}{10}$

$A = \frac{23}{10}$

$B = \frac{87}{10}$

$A = \frac{23}{10}$

$B = \frac{87}{10}$

$A = \frac{23}{10}$

$B = \frac{87}{10}$

Step 3.5

List all of the solutions.

$A = \frac{23}{10}, B = \frac{87}{10}$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x - 3} + \frac{B}{x + 7}$ with the values found for $A$ and $B$ .

$\frac{\frac{23}{10}}{x - 3} + \frac{\frac{87}{10}}{x + 7}$