Algebra Examples

$\frac{x^{2} - 1}{x^{3} + x^{2} + x}$

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

Rewrite $1$ as $1^{2}$ .

$\frac{x^{2} - 1^{2}}{x^{3} + x^{2} + x}$

Step 1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Raise $x$ to the power of $1$ .

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

Step 1.1.3.4

Factor $x$ out of $x^{1}$ .

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

Step 1.1.3.5

Factor $x$ out of $x \cdot x^{2} + x \cdot x$ .

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

Step 1.1.3.6

Factor $x$ out of $x (x^{2} + x) + x \cdot 1$ .

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

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 is 2nd order, $2$ terms are required in the numerator. The number of terms required in the numerator is always equal to the order of the factor in the denominator.

$\frac{A}{x} + \frac{B x + C}{x^{2} + x + 1}$

Step 1.3

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

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

Step 1.4

Reduce the expression by cancelling the common factors.

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Step 1.4.1

Cancel the common factor of $x$ .

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Step 1.4.1.1

Cancel the common factor.

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

Step 1.4.1.2

Rewrite the expression.

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

Step 1.4.2

Cancel the common factor of $x^{2} + x + 1$ .

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Step 1.4.2.1

Cancel the common factor.

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

Step 1.4.2.2

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

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

Step 1.5

Expand $(x + 1) (x - 1)$ using the FOIL Method.

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Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Apply the distributive property.

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

Step 1.5.3

Apply the distributive property.

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

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Multiply $x$ by $x$ .

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

Step 1.6.1.2

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

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

Step 1.6.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 1.6.1.4

Multiply $x$ by $1$ .

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

Step 1.6.1.5

Multiply $- 1$ by $1$ .

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

Step 1.6.2

Add $- x$ and $x$ .

$x^{2} + 0 - 1 = \frac{A (x (x^{2} + x + 1))}{x} + \frac{(B x + C) (x (x^{2} + x + 1))}{x^{2} + x + 1}$

Step 1.6.3

Add $x^{2}$ and $0$ .

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

Step 1.7

Simplify each term.

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Step 1.7.1

Cancel the common factor of $x$ .

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Step 1.7.1.1

Cancel the common factor.

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

Step 1.7.1.2

Divide $A (x^{2} + x + 1)$ by $1$ .

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

Step 1.7.2

Apply the distributive property.

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

Step 1.7.3

Multiply $A$ by $1$ .

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

Step 1.7.4

Cancel the common factor of $x^{2} + x + 1$ .

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Step 1.7.4.1

Cancel the common factor.

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

Step 1.7.4.2

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

$x^{2} - 1 = A x^{2} + A x + A + (B x + C) (x)$

Step 1.7.5

Apply the distributive property.

$x^{2} - 1 = A x^{2} + A x + A + B x \cdot x + C x$

Step 1.7.6

Multiply $x$ by $x$ by adding the exponents.

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Step 1.7.6.1

Move $x$ .

$x^{2} - 1 = A x^{2} + A x + A + B (x \cdot x) + C x$

Step 1.7.6.2

Multiply $x$ by $x$ .

$x^{2} - 1 = A x^{2} + A x + A + B x^{2} + C x$

Step 1.8

Simplify the expression.

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Step 1.8.1

Reorder $B$ and $x^{2}$ .

$x^{2} - 1 = A x^{2} + A x + A + B x^{2} + C x$

Step 1.8.2

Move $A$ .

$x^{2} - 1 = A x^{2} + A x + B x^{2} + C x + A$

Step 1.8.3

Move $A x$ .

$x^{2} - 1 = A x^{2} + B x^{2} + A x + C x + A$

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 $x^{2}$ 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 $x$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = A + C$

Step 2.3

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.

$- 1 = A$

Step 2.4

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

$1 = A + B$

$0 = A + C$

$- 1 = A$

$1 = A + B$

$0 = A + C$

$- 1 = A$

Step 3

Solve the system of equations.

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Step 3.1

Rewrite the equation as $A = - 1$ .

$A = - 1$

$1 = A + B$

$0 = A + C$

Step 3.2

Replace all occurrences of $A$ with $- 1$ in each equation.

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Step 3.2.1

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

$1 = (- 1) + B$

$A = - 1$

$0 = A + C$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Remove parentheses.

$1 = - 1 + B$

$A = - 1$

$0 = A + C$

$1 = - 1 + B$

$A = - 1$

$0 = A + C$

Step 3.2.3

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

$0 = (- 1) + C$

$1 = - 1 + B$

$A = - 1$

Step 3.2.4

Simplify the right side.

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Step 3.2.4.1

Remove parentheses.

$0 = - 1 + C$

$1 = - 1 + B$

$A = - 1$

$0 = - 1 + C$

$1 = - 1 + B$

$A = - 1$

$0 = - 1 + C$

$1 = - 1 + B$

$A = - 1$

Step 3.3

Solve for $C$ in $0 = - 1 + C$ .

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Step 3.3.1

Rewrite the equation as $- 1 + C = 0$ .

$- 1 + C = 0$

$1 = - 1 + B$

$A = - 1$

Step 3.3.2

Add $1$ to both sides of the equation.

$C = 1$

$1 = - 1 + B$

$A = - 1$

$C = 1$

$1 = - 1 + B$

$A = - 1$

Step 3.4

Replace all occurrences of $C$ with $1$ in each equation.

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Step 3.4.1

Rewrite the equation as $- 1 + B = 1$ .

$- 1 + B = 1$

$C = 1$

$A = - 1$

Step 3.4.2

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

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Step 3.4.2.1

Add $1$ to both sides of the equation.

$B = 1 + 1$

$C = 1$

$A = - 1$

Step 3.4.2.2

Add $1$ and $1$ .

$B = 2$

$C = 1$

$A = - 1$

$B = 2$

$C = 1$

$A = - 1$

$B = 2$

$C = 1$

$A = - 1$

Step 3.5

List all of the solutions.

$B = 2, C = 1, A = - 1$

Step 4

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

$- \frac{1}{x} + \frac{2 x + 1}{x^{2} + x + 1}$

Step 5

Multiply $2$ by $x$ .

$\frac{- 1}{x} + \frac{2 x + 1}{x^{2} + x + 1}$