Precalculus Examples

$\frac{x}{x^{2} - 16 x + 63}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

Factor $x^{2} - 16 x + 63$ using the AC method.

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Step 1.1.1

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 $63$ and whose sum is $- 16$ .

$- 9, - 7$

Step 1.1.2

Write the factored form using these integers.

$\frac{x}{(x - 9) (x - 7)}$

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 $A$ .

$\frac{A}{x - 9}$

Step 1.3

$\frac{A}{x - 9} + \frac{B}{x - 7}$

Step 1.4

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(x - 9) (x - 7)$ .

$\frac{x (x - 9) (x - 7)}{(x - 9) (x - 7)} = \frac{(A) (x - 9) (x - 7)}{x - 9} + \frac{(B) (x - 9) (x - 7)}{x - 7}$

Step 1.5

Cancel the common factor of $x - 9$ .

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Step 1.5.1

Cancel the common factor.

$\frac{x (x - 9) (x - 7)}{(x - 9) (x - 7)} = \frac{(A) (x - 9) (x - 7)}{x - 9} + \frac{(B) (x - 9) (x - 7)}{x - 7}$

Step 1.5.2

Rewrite the expression.

$\frac{x (x - 7)}{x - 7} = \frac{(A) (x - 9) (x - 7)}{x - 9} + \frac{(B) (x - 9) (x - 7)}{x - 7}$

Step 1.6

Cancel the common factor of $x - 7$ .

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Step 1.6.1

Cancel the common factor.

$\frac{x (x - 7)}{x - 7} = \frac{(A) (x - 9) (x - 7)}{x - 9} + \frac{(B) (x - 9) (x - 7)}{x - 7}$

Step 1.6.2

Divide $x$ by $1$ .

$x = \frac{(A) (x - 9) (x - 7)}{x - 9} + \frac{(B) (x - 9) (x - 7)}{x - 7}$

Step 1.7

Simplify each term.

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Step 1.7.1

Cancel the common factor of $x - 9$ .

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Step 1.7.1.1

Cancel the common factor.

$x = \frac{A (x - 9) (x - 7)}{x - 9} + \frac{(B) (x - 9) (x - 7)}{x - 7}$

Step 1.7.1.2

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

$x = (A) (x - 7) + \frac{(B) (x - 9) (x - 7)}{x - 7}$

Step 1.7.2

Apply the distributive property.

$x = A x + A \cdot - 7 + \frac{(B) (x - 9) (x - 7)}{x - 7}$

Step 1.7.3

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

$x = A x - 7 \cdot A + \frac{(B) (x - 9) (x - 7)}{x - 7}$

Step 1.7.4

Cancel the common factor of $x - 7$ .

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Step 1.7.4.1

Cancel the common factor.

$x = A x - 7 A + \frac{(B) (x - 9) (x - 7)}{x - 7}$

Step 1.7.4.2

Divide $(B) (x - 9)$ by $1$ .

$x = A x - 7 A + (B) (x - 9)$

Step 1.7.5

Apply the distributive property.

$x = A x - 7 A + B x + B \cdot - 9$

Step 1.7.6

Move $- 9$ to the left of $B$ .

$x = A x - 7 A + B x - 9 B$

Step 1.8

Move $- 7 A$ .

$x = A x + B x - 7 A - 9 B$

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.

$1 = A + B$

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.

$0 = - 7 A - 9 B$

Step 2.3

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

$1 = A + B$

$0 = - 7 A - 9 B$

$1 = A + B$

$0 = - 7 A - 9 B$

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 $A + B = 1$ .

$A + B = 1$

$0 = - 7 A - 9 B$

Step 3.1.2

Subtract $B$ from both sides of the equation.

$A = 1 - B$

$0 = - 7 A - 9 B$

$A = 1 - B$

$0 = - 7 A - 9 B$

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 $0 = - 7 A - 9 B$ with $1 - B$ .

$0 = - 7 (1 - B) - 9 B$

$A = 1 - B$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $- 7 (1 - B) - 9 B$ .

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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.

$0 = - 7 \cdot 1 - 7 (- B) - 9 B$

$A = 1 - B$

Step 3.2.2.1.1.2

Multiply $- 7$ by $1$ .

$0 = - 7 - 7 (- B) - 9 B$

$A = 1 - B$

Step 3.2.2.1.1.3

Multiply $- 1$ by $- 7$ .

$0 = - 7 + 7 B - 9 B$

$A = 1 - B$

$0 = - 7 + 7 B - 9 B$

$A = 1 - B$

Step 3.2.2.1.2

Subtract $9 B$ from $7 B$ .

$0 = - 7 - 2 B$

$A = 1 - B$

$0 = - 7 - 2 B$

$A = 1 - B$

$0 = - 7 - 2 B$

$A = 1 - B$

$0 = - 7 - 2 B$

$A = 1 - B$

Step 3.3

Solve for $B$ in $0 = - 7 - 2 B$ .

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Step 3.3.1

Rewrite the equation as $- 7 - 2 B = 0$ .

$- 7 - 2 B = 0$

$A = 1 - B$

Step 3.3.2

Add $7$ to both sides of the equation.

$- 2 B = 7$

$A = 1 - B$

Step 3.3.3

Divide each term in $- 2 B = 7$ by $- 2$ and simplify.

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Step 3.3.3.1

Divide each term in $- 2 B = 7$ by $- 2$ .

$\frac{- 2 B}{- 2} = \frac{7}{- 2}$

$A = 1 - B$

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.

$\frac{- 2 B}{- 2} = \frac{7}{- 2}$

$A = 1 - B$

Step 3.3.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{7}{- 2}$

$A = 1 - B$

$B = \frac{7}{- 2}$

$A = 1 - B$

$B = \frac{7}{- 2}$

$A = 1 - B$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Move the negative in front of the fraction.

$B = - \frac{7}{2}$

$A = 1 - B$

$B = - \frac{7}{2}$

$A = 1 - B$

$B = - \frac{7}{2}$

$A = 1 - B$

$B = - \frac{7}{2}$

$A = 1 - B$

Step 3.4

Replace all occurrences of $B$ with $- \frac{7}{2}$ in each equation.

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Step 3.4.1

Replace all occurrences of $B$ in $A = 1 - B$ with $- \frac{7}{2}$ .

$A = 1 - (- \frac{7}{2})$

$B = - \frac{7}{2}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $1 - (- \frac{7}{2})$ .

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Step 3.4.2.1.1

Multiply $- (- \frac{7}{2})$ .

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Step 3.4.2.1.1.1

Multiply $- 1$ by $- 1$ .

$A = 1 + 1 (\frac{7}{2})$

$B = - \frac{7}{2}$

Step 3.4.2.1.1.2

Multiply $\frac{7}{2}$ by $1$ .

$A = 1 + \frac{7}{2}$

$B = - \frac{7}{2}$

$A = 1 + \frac{7}{2}$

$B = - \frac{7}{2}$

Step 3.4.2.1.2

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

$A = \frac{2}{2} + \frac{7}{2}$

$B = - \frac{7}{2}$

Step 3.4.2.1.3

Combine the numerators over the common denominator.

$A = \frac{2 + 7}{2}$

$B = - \frac{7}{2}$

Step 3.4.2.1.4

Add $2$ and $7$ .

$A = \frac{9}{2}$

$B = - \frac{7}{2}$

$A = \frac{9}{2}$

$B = - \frac{7}{2}$

$A = \frac{9}{2}$

$B = - \frac{7}{2}$

$A = \frac{9}{2}$

$B = - \frac{7}{2}$

Step 3.5

List all of the solutions.

$A = \frac{9}{2}, B = - \frac{7}{2}$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x - 9} + \frac{B}{x - 7}$ with the values found for $A$ and $B$ .

$\frac{\frac{9}{2}}{x - 9} + \frac{- \frac{7}{2}}{x - 7}$