Precalculus Examples

$\frac{- 3 x^{2} + 24 x + 21}{x^{2} - 8 x + 7}$

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 $- 3 x^{2} + 24 x + 21$ .

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Step 1.1.1.1

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

$\frac{3 (- x^{2}) + 24 x + 21}{x^{2} - 8 x + 7}$

Step 1.1.1.2

Factor $3$ out of $24 x$ .

$\frac{3 (- x^{2}) + 3 (8 x) + 21}{x^{2} - 8 x + 7}$

Step 1.1.1.3

Factor $3$ out of $21$ .

$\frac{3 (- x^{2}) + 3 (8 x) + 3 (7)}{x^{2} - 8 x + 7}$

Step 1.1.1.4

Factor $3$ out of $3 (- x^{2}) + 3 (8 x)$ .

$\frac{3 (- x^{2} + 8 x) + 3 (7)}{x^{2} - 8 x + 7}$

Step 1.1.1.5

Factor $3$ out of $3 (- x^{2} + 8 x) + 3 (7)$ .

$\frac{3 (- x^{2} + 8 x + 7)}{x^{2} - 8 x + 7}$

Step 1.1.2

Factor $x^{2} - 8 x + 7$ 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 $7$ and whose sum is $- 8$ .

$- 7, - 1$

Step 1.1.2.2

Write the factored form using these integers.

$\frac{3 (- x^{2} + 8 x + 7)}{(x - 7) (x - 1)}$

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 $A$ .

$\frac{A}{x - 7}$

Step 1.3

$\frac{A}{x - 7} + \frac{B}{x - 1}$

Step 1.4

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(x - 7) (x - 1)$ .

$\frac{3 (- x^{2} + 8 x + 7) (x - 7) (x - 1)}{(x - 7) (x - 1)} = \frac{(A) (x - 7) (x - 1)}{x - 7} + \frac{(B) (x - 7) (x - 1)}{x - 1}$

Step 1.5

Cancel the common factor of $x - 7$ .

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Step 1.5.1

Cancel the common factor.

$\frac{3 (- x^{2} + 8 x + 7) (x - 7) (x - 1)}{(x - 7) (x - 1)} = \frac{(A) (x - 7) (x - 1)}{x - 7} + \frac{(B) (x - 7) (x - 1)}{x - 1}$

Step 1.5.2

Rewrite the expression.

$\frac{3 (- x^{2} + 8 x + 7) (x - 1)}{x - 1} = \frac{(A) (x - 7) (x - 1)}{x - 7} + \frac{(B) (x - 7) (x - 1)}{x - 1}$

Step 1.6

Cancel the common factor of $x - 1$ .

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Step 1.6.1

Cancel the common factor.

$\frac{3 (- x^{2} + 8 x + 7) (x - 1)}{x - 1} = \frac{(A) (x - 7) (x - 1)}{x - 7} + \frac{(B) (x - 7) (x - 1)}{x - 1}$

Step 1.6.2

Divide $3 (- x^{2} + 8 x + 7)$ by $1$ .

$3 (- x^{2} + 8 x + 7) = \frac{(A) (x - 7) (x - 1)}{x - 7} + \frac{(B) (x - 7) (x - 1)}{x - 1}$

Step 1.7

Apply the distributive property.

$3 (- x^{2}) + 3 (8 x) + 3 \cdot 7 = \frac{(A) (x - 7) (x - 1)}{x - 7} + \frac{(B) (x - 7) (x - 1)}{x - 1}$

Step 1.8

Simplify.

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Step 1.8.1

Multiply $- 1$ by $3$ .

$- 3 x^{2} + 3 (8 x) + 3 \cdot 7 = \frac{(A) (x - 7) (x - 1)}{x - 7} + \frac{(B) (x - 7) (x - 1)}{x - 1}$

Step 1.8.2

Multiply $8$ by $3$ .

$- 3 x^{2} + 24 x + 3 \cdot 7 = \frac{(A) (x - 7) (x - 1)}{x - 7} + \frac{(B) (x - 7) (x - 1)}{x - 1}$

Step 1.8.3

Multiply $3$ by $7$ .

$- 3 x^{2} + 24 x + 21 = \frac{(A) (x - 7) (x - 1)}{x - 7} + \frac{(B) (x - 7) (x - 1)}{x - 1}$

Step 1.9

Simplify each term.

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Step 1.9.1

Cancel the common factor of $x - 7$ .

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Step 1.9.1.1

Cancel the common factor.

$- 3 x^{2} + 24 x + 21 = \frac{A (x - 7) (x - 1)}{x - 7} + \frac{(B) (x - 7) (x - 1)}{x - 1}$

Step 1.9.1.2

Divide $(A) (x - 1)$ by $1$ .

$- 3 x^{2} + 24 x + 21 = (A) (x - 1) + \frac{(B) (x - 7) (x - 1)}{x - 1}$

Step 1.9.2

Apply the distributive property.

$- 3 x^{2} + 24 x + 21 = A x + A \cdot - 1 + \frac{(B) (x - 7) (x - 1)}{x - 1}$

Step 1.9.3

Move $- 1$ to the left of $A$ .

$- 3 x^{2} + 24 x + 21 = A x - 1 \cdot A + \frac{(B) (x - 7) (x - 1)}{x - 1}$

Step 1.9.4

Rewrite $- 1 A$ as $- A$ .

$- 3 x^{2} + 24 x + 21 = A x - A + \frac{(B) (x - 7) (x - 1)}{x - 1}$

Step 1.9.5

Cancel the common factor of $x - 1$ .

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Step 1.9.5.1

Cancel the common factor.

$- 3 x^{2} + 24 x + 21 = A x - A + \frac{(B) (x - 7) (x - 1)}{x - 1}$

Step 1.9.5.2

Divide $(B) (x - 7)$ by $1$ .

$- 3 x^{2} + 24 x + 21 = A x - A + (B) (x - 7)$

Step 1.9.6

Apply the distributive property.

$- 3 x^{2} + 24 x + 21 = A x - A + B x + B \cdot - 7$

Step 1.9.7

Move $- 7$ to the left of $B$ .

$- 3 x^{2} + 24 x + 21 = A x - A + B x - 7 B$

Step 1.10

Move $- A$ .

$- 3 x^{2} + 24 x + 21 = A x + B x - A - 7 B$

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.

$24 = A + B$

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.

$21 = - 1 A - 7 B$

Step 2.3

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

$24 = A + B$

$21 = - 1 A - 7 B$

$24 = A + B$

$21 = - 1 A - 7 B$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $A$ in $24 = A + B$ .

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Step 3.1.1

Rewrite the equation as $A + B = 24$ .

$A + B = 24$

$21 = - 1 A - 7 B$

Step 3.1.2

Subtract $B$ from both sides of the equation.

$A = 24 - B$

$21 = - 1 A - 7 B$

$A = 24 - B$

$21 = - 1 A - 7 B$

Step 3.2

Replace all occurrences of $A$ with $24 - B$ in each equation.

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Step 3.2.1

Replace all occurrences of $A$ in $21 = - 1 A - 7 B$ with $24 - B$ .

$21 = - 1 (24 - B) - 7 B$

$A = 24 - B$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $- 1 (24 - B) - 7 B$ .

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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.

$21 = - 1 \cdot 24 - 1 (- B) - 7 B$

$A = 24 - B$

Step 3.2.2.1.1.2

Multiply $- 1$ by $24$ .

$21 = - 24 - 1 (- B) - 7 B$

$A = 24 - B$

Step 3.2.2.1.1.3

Multiply $- 1 (- B)$ .

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Step 3.2.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$21 = - 24 + 1 B - 7 B$

$A = 24 - B$

Step 3.2.2.1.1.3.2

Multiply $B$ by $1$ .

$21 = - 24 + B - 7 B$

$A = 24 - B$

$21 = - 24 + B - 7 B$

$A = 24 - B$

$21 = - 24 + B - 7 B$

$A = 24 - B$

Step 3.2.2.1.2

Subtract $7 B$ from $B$ .

$21 = - 24 - 6 B$

$A = 24 - B$

$21 = - 24 - 6 B$

$A = 24 - B$

$21 = - 24 - 6 B$

$A = 24 - B$

$21 = - 24 - 6 B$

$A = 24 - B$

Step 3.3

Solve for $B$ in $21 = - 24 - 6 B$ .

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Step 3.3.1

Rewrite the equation as $- 24 - 6 B = 21$ .

$- 24 - 6 B = 21$

$A = 24 - B$

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 $24$ to both sides of the equation.

$- 6 B = 21 + 24$

$A = 24 - B$

Step 3.3.2.2

Add $21$ and $24$ .

$- 6 B = 45$

$A = 24 - B$

$- 6 B = 45$

$A = 24 - B$

Step 3.3.3

Divide each term in $- 6 B = 45$ by $- 6$ and simplify.

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Step 3.3.3.1

Divide each term in $- 6 B = 45$ by $- 6$ .

$\frac{- 6 B}{- 6} = \frac{45}{- 6}$

$A = 24 - B$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 6$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 6 B}{- 6} = \frac{45}{- 6}$

$A = 24 - B$

Step 3.3.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{45}{- 6}$

$A = 24 - B$

$B = \frac{45}{- 6}$

$A = 24 - B$

$B = \frac{45}{- 6}$

$A = 24 - B$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Cancel the common factor of $45$ and $- 6$ .

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Step 3.3.3.3.1.1

Factor $3$ out of $45$ .

$B = \frac{3 (15)}{- 6}$

$A = 24 - B$

Step 3.3.3.3.1.2

Cancel the common factors.

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Step 3.3.3.3.1.2.1

Factor $3$ out of $- 6$ .

$B = \frac{3 \cdot 15}{3 \cdot - 2}$

$A = 24 - B$

Step 3.3.3.3.1.2.2

Cancel the common factor.

$B = \frac{3 \cdot 15}{3 \cdot - 2}$

$A = 24 - B$

Step 3.3.3.3.1.2.3

Rewrite the expression.

$B = \frac{15}{- 2}$

$A = 24 - B$

$B = \frac{15}{- 2}$

$A = 24 - B$

$B = \frac{15}{- 2}$

$A = 24 - B$

Step 3.3.3.3.2

Move the negative in front of the fraction.

$B = - \frac{15}{2}$

$A = 24 - B$

$B = - \frac{15}{2}$

$A = 24 - B$

$B = - \frac{15}{2}$

$A = 24 - B$

$B = - \frac{15}{2}$

$A = 24 - B$

Step 3.4

Replace all occurrences of $B$ with $- \frac{15}{2}$ in each equation.

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Step 3.4.1

Replace all occurrences of $B$ in $A = 24 - B$ with $- \frac{15}{2}$ .

$A = 24 - (- \frac{15}{2})$

$B = - \frac{15}{2}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $24 - (- \frac{15}{2})$ .

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Step 3.4.2.1.1

Multiply $- (- \frac{15}{2})$ .

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Step 3.4.2.1.1.1

Multiply $- 1$ by $- 1$ .

$A = 24 + 1 (\frac{15}{2})$

$B = - \frac{15}{2}$

Step 3.4.2.1.1.2

Multiply $\frac{15}{2}$ by $1$ .

$A = 24 + \frac{15}{2}$

$B = - \frac{15}{2}$

$A = 24 + \frac{15}{2}$

$B = - \frac{15}{2}$

Step 3.4.2.1.2

To write $24$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$A = 24 \cdot \frac{2}{2} + \frac{15}{2}$

$B = - \frac{15}{2}$

Step 3.4.2.1.3

Combine $24$ and $\frac{2}{2}$ .

$A = \frac{24 \cdot 2}{2} + \frac{15}{2}$

$B = - \frac{15}{2}$

Step 3.4.2.1.4

Combine the numerators over the common denominator.

$A = \frac{24 \cdot 2 + 15}{2}$

$B = - \frac{15}{2}$

Step 3.4.2.1.5

Simplify the numerator.

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Step 3.4.2.1.5.1

Multiply $24$ by $2$ .

$A = \frac{48 + 15}{2}$

$B = - \frac{15}{2}$

Step 3.4.2.1.5.2

Add $48$ and $15$ .

$A = \frac{63}{2}$

$B = - \frac{15}{2}$

$A = \frac{63}{2}$

$B = - \frac{15}{2}$

$A = \frac{63}{2}$

$B = - \frac{15}{2}$

$A = \frac{63}{2}$

$B = - \frac{15}{2}$

$A = \frac{63}{2}$

$B = - \frac{15}{2}$

Step 3.5

List all of the solutions.

$A = \frac{63}{2}, B = - \frac{15}{2}$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x - 7} + \frac{B}{x - 1}$ with the values found for $A$ and $B$ .

$\frac{\frac{63}{2}}{x - 7} + \frac{- \frac{15}{2}}{x - 1}$