Precalculus Examples

$\frac{11}{x (1 - x)}$

Step 1

Decompose the fraction and multiply through by the common denominator.

Tap for more steps...

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 $B$ .

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

Step 1.2

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

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

Step 1.3

Cancel the common factor of $x$ .

Tap for more steps...

Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 1.4

Cancel the common factor of $1 - x$ .

Tap for more steps...

Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Divide $11$ by $1$ .

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

Step 1.5

Simplify each term.

Tap for more steps...

Step 1.5.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 1.5.1.1

Cancel the common factor.

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

Step 1.5.1.2

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

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

Step 1.5.2

Apply the distributive property.

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

Step 1.5.3

Multiply $A$ by $1$ .

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

Step 1.5.4

Rewrite using the commutative property of multiplication.

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

Step 1.5.5

Cancel the common factor of $1 - x$ .

Tap for more steps...

Step 1.5.5.1

Cancel the common factor.

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

Step 1.5.5.2

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

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

$11 = A - A x + B x$

Step 1.6

Simplify the expression.

Tap for more steps...

Step 1.6.1

Move $A$ .

$11 = A - x A + B x$

Step 1.6.2

Reorder $B$ and $x$ .

$11 = A - x A + B x$

Step 1.6.3

Move $A$ .

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

Step 2

Create equations for the partial fraction variables and use them to set up a system of equations.

Tap for more steps...

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.

$0 = - 1 A + B$

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.

$11 = A$

Step 2.3

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

$0 = - 1 A + B$

$11 = A$

$0 = - 1 A + B$

$11 = A$

Step 3

Solve the system of equations.

Tap for more steps...

Step 3.1

Rewrite the equation as $A = 11$ .

$A = 11$

$0 = - 1 A + B$

Step 3.2

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

Tap for more steps...

Step 3.2.1

Replace all occurrences of $A$ in $0 = - 1 A + B$ with $11$ .

$0 = - 1 \cdot 11 + B$

$A = 11$

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

Multiply $- 1$ by $11$ .

$0 = - 11 + B$

$A = 11$

$0 = - 11 + B$

$A = 11$

$0 = - 11 + B$

$A = 11$

Step 3.3

Solve for $B$ in $0 = - 11 + B$ .

Tap for more steps...

Step 3.3.1

Rewrite the equation as $- 11 + B = 0$ .

$- 11 + B = 0$

$A = 11$

Step 3.3.2

Add $11$ to both sides of the equation.

$B = 11$

$A = 11$

$B = 11$

$A = 11$

Step 3.4

Solve the system of equations.

$B = 11 A = 11$

Step 3.5

List all of the solutions.

$B = 11, A = 11$

Step 4

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

$\frac{11}{x} + \frac{11}{1 - x}$