Precalculus Examples

$\frac{6 x - 11}{{(x - 1)}^{2}}$

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 $A$ .

$\frac{A}{{(x - 1)}^{2}}$

Step 1.2

$\frac{A}{{(x - 1)}^{2}} + \frac{B}{x - 1}$

Step 1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is ${(x - 1)}^{2}$ .

$\frac{(6 x - 11) {(x - 1)}^{2}}{{(x - 1)}^{2}} = \frac{(A) {(x - 1)}^{2}}{{(x - 1)}^{2}} + \frac{(B) {(x - 1)}^{2}}{x - 1}$

Step 1.4

Cancel the common factor of ${(x - 1)}^{2}$ .

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Step 1.4.1

Cancel the common factor.

$\frac{(6 x - 11) {(x - 1)}^{2}}{{(x - 1)}^{2}} = \frac{(A) {(x - 1)}^{2}}{{(x - 1)}^{2}} + \frac{(B) {(x - 1)}^{2}}{x - 1}$

Step 1.4.2

Divide $6 x - 11$ by $1$ .

$6 x - 11 = \frac{(A) {(x - 1)}^{2}}{{(x - 1)}^{2}} + \frac{(B) {(x - 1)}^{2}}{x - 1}$

Step 1.5

Simplify each term.

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Step 1.5.1

Cancel the common factor of ${(x - 1)}^{2}$ .

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Step 1.5.1.1

Cancel the common factor.

$6 x - 11 = \frac{A {(x - 1)}^{2}}{{(x - 1)}^{2}} + \frac{(B) {(x - 1)}^{2}}{x - 1}$

Step 1.5.1.2

Divide $A$ by $1$ .

$6 x - 11 = A + \frac{(B) {(x - 1)}^{2}}{x - 1}$

Step 1.5.2

Cancel the common factor of ${(x - 1)}^{2}$ and $x - 1$ .

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Step 1.5.2.1

Factor $x - 1$ out of $(B) {(x - 1)}^{2}$ .

$6 x - 11 = A + \frac{(x - 1) (B (x - 1))}{x - 1}$

Step 1.5.2.2

Cancel the common factors.

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Step 1.5.2.2.1

Multiply by $1$ .

$6 x - 11 = A + \frac{(x - 1) (B (x - 1))}{(x - 1) \cdot 1}$

Step 1.5.2.2.2

Cancel the common factor.

$6 x - 11 = A + \frac{(x - 1) (B (x - 1))}{(x - 1) \cdot 1}$

Step 1.5.2.2.3

Rewrite the expression.

$6 x - 11 = A + \frac{B (x - 1)}{1}$

Step 1.5.2.2.4

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

$6 x - 11 = A + B (x - 1)$

Step 1.5.3

Apply the distributive property.

$6 x - 11 = A + B x + B \cdot - 1$

Step 1.5.4

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

$6 x - 11 = A + B x - 1 \cdot B$

Step 1.5.5

Rewrite $- 1 B$ as $- B$ .

$6 x - 11 = A + B x - B$

Step 1.6

Reorder $A$ and $B x$ .

$6 x - 11 = B x + A - B$

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.

$6 = B$

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.

$- 11 = A - 1 B$

Step 2.3

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

$6 = B$

$- 11 = A - 1 B$

$6 = B$

$- 11 = A - 1 B$

Step 3

Solve the system of equations.

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Step 3.1

Rewrite the equation as $B = 6$ .

$B = 6$

$- 11 = A - 1 B$

Step 3.2

Replace all occurrences of $B$ with $6$ in each equation.

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Step 3.2.1

Replace all occurrences of $B$ in $- 11 = A - 1 B$ with $6$ .

$- 11 = A - 1 \cdot 6$

$B = 6$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- 1$ by $6$ .

$- 11 = A - 6$

$B = 6$

$- 11 = A - 6$

$B = 6$

$- 11 = A - 6$

$B = 6$

Step 3.3

Solve for $A$ in $- 11 = A - 6$ .

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Step 3.3.1

Rewrite the equation as $A - 6 = - 11$ .

$A - 6 = - 11$

$B = 6$

Step 3.3.2

Move all terms not containing $A$ to the right side of the equation.

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Step 3.3.2.1

Add $6$ to both sides of the equation.

$A = - 11 + 6$

$B = 6$

Step 3.3.2.2

Add $- 11$ and $6$ .

$A = - 5$

$B = 6$

$A = - 5$

$B = 6$

$A = - 5$

$B = 6$

Step 3.4

Solve the system of equations.

$A = - 5 B = 6$

Step 3.5

List all of the solutions.

$A = - 5, B = 6$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{{(x - 1)}^{2}} + \frac{B}{x - 1}$ with the values found for $A$ and $B$ .

$\frac{- 5}{{(x - 1)}^{2}} + \frac{6}{x - 1}$