Precalculus Examples

$\frac{6 x^{2} + 12 x - 177}{x^{2} + x - 30}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

Factor the fraction.

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Step 1.1.1

Factor $3$ out of $6 x^{2} + 12 x - 177$ .

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Step 1.1.1.1

Factor $3$ out of $6 x^{2}$ .

$\frac{3 (2 x^{2}) + 12 x - 177}{x^{2} + x - 30}$

Step 1.1.1.2

Factor $3$ out of $12 x$ .

$\frac{3 (2 x^{2}) + 3 (4 x) - 177}{x^{2} + x - 30}$

Step 1.1.1.3

Factor $3$ out of $- 177$ .

$\frac{3 (2 x^{2}) + 3 (4 x) + 3 (- 59)}{x^{2} + x - 30}$

Step 1.1.1.4

Factor $3$ out of $3 (2 x^{2}) + 3 (4 x)$ .

$\frac{3 (2 x^{2} + 4 x) + 3 (- 59)}{x^{2} + x - 30}$

Step 1.1.1.5

Factor $3$ out of $3 (2 x^{2} + 4 x) + 3 (- 59)$ .

$\frac{3 (2 x^{2} + 4 x - 59)}{x^{2} + x - 30}$

Step 1.1.2

Factor $x^{2} + x - 30$ using the AC method.

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Step 1.1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 30$ and whose sum is $1$ .

$- 5, 6$

Step 1.1.2.2

Write the factored form using these integers.

$\frac{3 (2 x^{2} + 4 x - 59)}{(x - 5) (x + 6)}$

Step 1.2

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 $A$ .

$\frac{A}{x - 5}$

Step 1.3

$\frac{A}{x - 5} + \frac{B}{x + 6}$

Step 1.4

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(x - 5) (x + 6)$ .

$\frac{3 (2 x^{2} + 4 x - 59) (x - 5) (x + 6)}{(x - 5) (x + 6)} = \frac{(A) (x - 5) (x + 6)}{x - 5} + \frac{(B) (x - 5) (x + 6)}{x + 6}$

Step 1.5

Cancel the common factor of $x - 5$ .

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Step 1.5.1

Cancel the common factor.

$\frac{3 (2 x^{2} + 4 x - 59) (x - 5) (x + 6)}{(x - 5) (x + 6)} = \frac{(A) (x - 5) (x + 6)}{x - 5} + \frac{(B) (x - 5) (x + 6)}{x + 6}$

Step 1.5.2

Rewrite the expression.

$\frac{3 (2 x^{2} + 4 x - 59) (x + 6)}{x + 6} = \frac{(A) (x - 5) (x + 6)}{x - 5} + \frac{(B) (x - 5) (x + 6)}{x + 6}$

Step 1.6

Cancel the common factor of $x + 6$ .

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Step 1.6.1

Cancel the common factor.

$\frac{3 (2 x^{2} + 4 x - 59) (x + 6)}{x + 6} = \frac{(A) (x - 5) (x + 6)}{x - 5} + \frac{(B) (x - 5) (x + 6)}{x + 6}$

Step 1.6.2

Divide $3 (2 x^{2} + 4 x - 59)$ by $1$ .

$3 (2 x^{2} + 4 x - 59) = \frac{(A) (x - 5) (x + 6)}{x - 5} + \frac{(B) (x - 5) (x + 6)}{x + 6}$

Step 1.7

Apply the distributive property.

$3 (2 x^{2}) + 3 (4 x) + 3 \cdot - 59 = \frac{(A) (x - 5) (x + 6)}{x - 5} + \frac{(B) (x - 5) (x + 6)}{x + 6}$

Step 1.8

Simplify.

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Step 1.8.1

Multiply $2$ by $3$ .

$6 x^{2} + 3 (4 x) + 3 \cdot - 59 = \frac{(A) (x - 5) (x + 6)}{x - 5} + \frac{(B) (x - 5) (x + 6)}{x + 6}$

Step 1.8.2

Multiply $4$ by $3$ .

$6 x^{2} + 12 x + 3 \cdot - 59 = \frac{(A) (x - 5) (x + 6)}{x - 5} + \frac{(B) (x - 5) (x + 6)}{x + 6}$

Step 1.8.3

Multiply $3$ by $- 59$ .

$6 x^{2} + 12 x - 177 = \frac{(A) (x - 5) (x + 6)}{x - 5} + \frac{(B) (x - 5) (x + 6)}{x + 6}$

Step 1.9

Simplify each term.

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Step 1.9.1

Cancel the common factor of $x - 5$ .

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Step 1.9.1.1

Cancel the common factor.

$6 x^{2} + 12 x - 177 = \frac{A (x - 5) (x + 6)}{x - 5} + \frac{(B) (x - 5) (x + 6)}{x + 6}$

Step 1.9.1.2

Divide $(A) (x + 6)$ by $1$ .

$6 x^{2} + 12 x - 177 = (A) (x + 6) + \frac{(B) (x - 5) (x + 6)}{x + 6}$

Step 1.9.2

Apply the distributive property.

$6 x^{2} + 12 x - 177 = A x + A \cdot 6 + \frac{(B) (x - 5) (x + 6)}{x + 6}$

Step 1.9.3

Move $6$ to the left of $A$ .

$6 x^{2} + 12 x - 177 = A x + 6 \cdot A + \frac{(B) (x - 5) (x + 6)}{x + 6}$

Step 1.9.4

Cancel the common factor of $x + 6$ .

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Step 1.9.4.1

Cancel the common factor.

$6 x^{2} + 12 x - 177 = A x + 6 A + \frac{(B) (x - 5) (x + 6)}{x + 6}$

Step 1.9.4.2

Divide $(B) (x - 5)$ by $1$ .

$6 x^{2} + 12 x - 177 = A x + 6 A + (B) (x - 5)$

Step 1.9.5

Apply the distributive property.

$6 x^{2} + 12 x - 177 = A x + 6 A + B x + B \cdot - 5$

Step 1.9.6

Move $- 5$ to the left of $B$ .

$6 x^{2} + 12 x - 177 = A x + 6 A + B x - 5 B$

Step 1.10

Move $6 A$ .

$6 x^{2} + 12 x - 177 = A x + B x + 6 A - 5 B$

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.

$12 = A + B$

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.

$- 177 = 6 A - 5 B$

Step 2.3

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

$12 = A + B$

$- 177 = 6 A - 5 B$

$12 = A + B$

$- 177 = 6 A - 5 B$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $A$ in $12 = A + B$ .

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Step 3.1.1

Rewrite the equation as $A + B = 12$ .

$A + B = 12$

$- 177 = 6 A - 5 B$

Step 3.1.2

Subtract $B$ from both sides of the equation.

$A = 12 - B$

$- 177 = 6 A - 5 B$

$A = 12 - B$

$- 177 = 6 A - 5 B$

Step 3.2

Replace all occurrences of $A$ with $12 - B$ in each equation.

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Step 3.2.1

Replace all occurrences of $A$ in $- 177 = 6 A - 5 B$ with $12 - B$ .

$- 177 = 6 (12 - B) - 5 B$

$A = 12 - B$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $6 (12 - B) - 5 B$ .

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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.

$- 177 = 6 \cdot 12 + 6 (- B) - 5 B$

$A = 12 - B$

Step 3.2.2.1.1.2

Multiply $6$ by $12$ .

$- 177 = 72 + 6 (- B) - 5 B$

$A = 12 - B$

Step 3.2.2.1.1.3

Multiply $- 1$ by $6$ .

$- 177 = 72 - 6 B - 5 B$

$A = 12 - B$

$- 177 = 72 - 6 B - 5 B$

$A = 12 - B$

Step 3.2.2.1.2

Subtract $5 B$ from $- 6 B$ .

$- 177 = 72 - 11 B$

$A = 12 - B$

$- 177 = 72 - 11 B$

$A = 12 - B$

$- 177 = 72 - 11 B$

$A = 12 - B$

$- 177 = 72 - 11 B$

$A = 12 - B$

Step 3.3

Solve for $B$ in $- 177 = 72 - 11 B$ .

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Step 3.3.1

Rewrite the equation as $72 - 11 B = - 177$ .

$72 - 11 B = - 177$

$A = 12 - B$

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 $72$ from both sides of the equation.

$- 11 B = - 177 - 72$

$A = 12 - B$

Step 3.3.2.2

Subtract $72$ from $- 177$ .

$- 11 B = - 249$

$A = 12 - B$

$- 11 B = - 249$

$A = 12 - B$

Step 3.3.3

Divide each term in $- 11 B = - 249$ by $- 11$ and simplify.

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Step 3.3.3.1

Divide each term in $- 11 B = - 249$ by $- 11$ .

$\frac{- 11 B}{- 11} = \frac{- 249}{- 11}$

$A = 12 - B$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 11$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 11 B}{- 11} = \frac{- 249}{- 11}$

$A = 12 - B$

Step 3.3.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{- 249}{- 11}$

$A = 12 - B$

$B = \frac{- 249}{- 11}$

$A = 12 - B$

$B = \frac{- 249}{- 11}$

$A = 12 - B$

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.

$B = \frac{249}{11}$

$A = 12 - B$

$B = \frac{249}{11}$

$A = 12 - B$

$B = \frac{249}{11}$

$A = 12 - B$

$B = \frac{249}{11}$

$A = 12 - B$

Step 3.4

Replace all occurrences of $B$ with $\frac{249}{11}$ in each equation.

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Step 3.4.1

Replace all occurrences of $B$ in $A = 12 - B$ with $\frac{249}{11}$ .

$A = 12 - (\frac{249}{11})$

$B = \frac{249}{11}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $12 - (\frac{249}{11})$ .

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Step 3.4.2.1.1

To write $12$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$A = 12 \cdot \frac{11}{11} - \frac{249}{11}$

$B = \frac{249}{11}$

Step 3.4.2.1.2

Combine $12$ and $\frac{11}{11}$ .

$A = \frac{12 \cdot 11}{11} - \frac{249}{11}$

$B = \frac{249}{11}$

Step 3.4.2.1.3

Combine the numerators over the common denominator.

$A = \frac{12 \cdot 11 - 249}{11}$

$B = \frac{249}{11}$

Step 3.4.2.1.4

Simplify the numerator.

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Step 3.4.2.1.4.1

Multiply $12$ by $11$ .

$A = \frac{132 - 249}{11}$

$B = \frac{249}{11}$

Step 3.4.2.1.4.2

Subtract $249$ from $132$ .

$A = \frac{- 117}{11}$

$B = \frac{249}{11}$

$A = \frac{- 117}{11}$

$B = \frac{249}{11}$

Step 3.4.2.1.5

Move the negative in front of the fraction.

$A = - \frac{117}{11}$

$B = \frac{249}{11}$

$A = - \frac{117}{11}$

$B = \frac{249}{11}$

$A = - \frac{117}{11}$

$B = \frac{249}{11}$

$A = - \frac{117}{11}$

$B = \frac{249}{11}$

Step 3.5

List all of the solutions.

$A = - \frac{117}{11}, B = \frac{249}{11}$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x - 5} + \frac{B}{x + 6}$ with the values found for $A$ and $B$ .

$\frac{- \frac{117}{11}}{x - 5} + \frac{\frac{249}{11}}{x + 6}$