Calculus Examples

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

Step 1

Decompose the fraction and multiply through by the common denominator.

Tap for more steps...

Step 1.1

Factor the fraction.

Tap for more steps...

Step 1.1.1

Rewrite $1$ as $1^{3}$ .

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

Step 1.1.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 1$ .

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

Step 1.1.3

Simplify.

Tap for more steps...

Step 1.1.3.1

Multiply $x$ by $1$ .

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

Step 1.1.3.2

One to any power is one.

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

Step 1.1.4

Rewrite $1$ as $1^{3}$ .

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

Step 1.1.5

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 1$ .

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

Step 1.1.6

Factor.

Tap for more steps...

Step 1.1.6.1

Simplify.

Tap for more steps...

Step 1.1.6.1.1

Multiply $- 1$ by $1$ .

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

Step 1.1.6.1.2

One to any power is one.

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

Step 1.1.6.2

Remove unnecessary parentheses.

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

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 in the denominator is linear, put a single variable in its place $B$ .

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

Step 1.3

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.

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

Step 1.4

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

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

Step 1.5

Reduce the expression by cancelling the common factors.

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.

Step 1.5.1.2

Rewrite the expression.

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

Step 1.5.2

Cancel the common factor of $x + 1$ .

Tap for more steps...

Step 1.5.2.1

Cancel the common factor.

Step 1.5.2.2

Rewrite the expression.

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

Step 1.5.3

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

Tap for more steps...

Step 1.5.3.1

Cancel the common factor.

Step 1.5.3.2

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

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

Step 1.6

Expand $(x - 1) (x^{2} + x + 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 1.7

Simplify terms.

Tap for more steps...

Step 1.7.1

Combine the opposite terms in $x \cdot x^{2} + x \cdot x + x \cdot 1 - 1 x^{2} - 1 x - 1 \cdot 1$ .

Tap for more steps...

Step 1.7.1.1

Reorder the factors in the terms $x \cdot 1$ and $- 1 x$ .

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

Step 1.7.1.2

Subtract $1 x$ from $1 x$ .

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

Step 1.7.1.3

Add $x \cdot x^{2} + x \cdot x - 1 x^{2}$ and $0$ .

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

Step 1.7.2

Simplify each term.

Tap for more steps...

Step 1.7.2.1

Multiply $x$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 1.7.2.1.1

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 1.7.2.1.1.1

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

Step 1.7.2.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.7.2.1.2

Add $1$ and $2$ .

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

Step 1.7.2.2

Multiply $x$ by $x$ .

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

Step 1.7.2.3

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

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

Step 1.7.2.4

Multiply $- 1$ by $1$ .

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

Step 1.7.3

Combine the opposite terms in $x^{3} + x^{2} - x^{2} - 1$ .

Tap for more steps...

Step 1.7.3.1

Subtract $x^{2}$ from $x^{2}$ .

$x^{3} + 0 - 1 = \frac{A (x (x + 1) (x^{2} - x + 1))}{x} + \frac{(B) (x (x + 1) (x^{2} - x + 1))}{x + 1} + \frac{(C x + D) (x (x + 1) (x^{2} - x + 1))}{x^{2} - x + 1}$

Step 1.7.3.2

Add $x^{3}$ and $0$ .

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

Step 1.8

Simplify each term.

Tap for more steps...

Step 1.8.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 1.8.1.1

Cancel the common factor.

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

Step 1.8.1.2

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

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

Step 1.8.2

Expand $(x + 1) (x^{2} - x + 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 1.8.3

Simplify each term.

Tap for more steps...

Step 1.8.3.1

Multiply $x$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 1.8.3.1.1

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 1.8.3.1.1.1

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

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

Step 1.8.3.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.8.3.1.2

Add $1$ and $2$ .

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

Step 1.8.3.2

Rewrite using the commutative property of multiplication.

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

Step 1.8.3.3

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

Tap for more steps...

Step 1.8.3.3.1

Move $x$ .

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

Step 1.8.3.3.2

Multiply $x$ by $x$ .

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

Step 1.8.3.4

Multiply $x$ by $1$ .

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

Step 1.8.3.5

Multiply $x^{2}$ by $1$ .

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

Step 1.8.3.6

Multiply $- x$ by $1$ .

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

Step 1.8.3.7

Multiply $1$ by $1$ .

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

Step 1.8.4

Combine the opposite terms in $x^{3} - x^{2} + x + x^{2} - x + 1$ .

Tap for more steps...

Step 1.8.4.1

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

$x^{3} - 1 = A (x^{3} + x + 0 - x + 1) + \frac{(B) (x (x + 1) (x^{2} - x + 1))}{x + 1} + \frac{(C x + D) (x (x + 1) (x^{2} - x + 1))}{x^{2} - x + 1}$

Step 1.8.4.2

Add $x^{3} + x$ and $0$ .

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

Step 1.8.4.3

Subtract $x$ from $x$ .

$x^{3} - 1 = A (x^{3} + 0 + 1) + \frac{(B) (x (x + 1) (x^{2} - x + 1))}{x + 1} + \frac{(C x + D) (x (x + 1) (x^{2} - x + 1))}{x^{2} - x + 1}$

Step 1.8.4.4

Add $x^{3}$ and $0$ .

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

Step 1.8.5

Apply the distributive property.

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

Step 1.8.6

Multiply $A$ by $1$ .

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

Step 1.8.7

Cancel the common factor of $x + 1$ .

Tap for more steps...

Step 1.8.7.1

Cancel the common factor.

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

Step 1.8.7.2

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

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

Step 1.8.8

Apply the distributive property.

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

Step 1.8.9

Simplify.

Tap for more steps...

Step 1.8.9.1

Multiply $x$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 1.8.9.1.1

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 1.8.9.1.1.1

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

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

Step 1.8.9.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.8.9.1.2

Add $1$ and $2$ .

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

Step 1.8.9.2

Rewrite using the commutative property of multiplication.

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

Step 1.8.9.3

Multiply $x$ by $1$ .

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

Step 1.8.10

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

Tap for more steps...

Step 1.8.10.1

Move $x$ .

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

Step 1.8.10.2

Multiply $x$ by $x$ .

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

Step 1.8.11

Apply the distributive property.

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

Step 1.8.12

Rewrite using the commutative property of multiplication.

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

Step 1.8.13

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

Tap for more steps...

Step 1.8.13.1

Cancel the common factor.

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

Step 1.8.13.2

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

$x^{3} - 1 = A x^{3} + A + B x^{3} - B x^{2} + B x + (C x + D) (x (x + 1))$

Step 1.8.14

Apply the distributive property.

$x^{3} - 1 = A x^{3} + A + B x^{3} - B x^{2} + B x + (C x + D) (x \cdot x + x \cdot 1)$

Step 1.8.15

Multiply $x$ by $x$ .

$x^{3} - 1 = A x^{3} + A + B x^{3} - B x^{2} + B x + (C x + D) (x^{2} + x \cdot 1)$

Step 1.8.16

Multiply $x$ by $1$ .

$x^{3} - 1 = A x^{3} + A + B x^{3} - B x^{2} + B x + (C x + D) (x^{2} + x)$

Step 1.8.17

Expand $(C x + D) (x^{2} + x)$ using the FOIL Method.

Tap for more steps...

Step 1.8.17.1

Apply the distributive property.

$x^{3} - 1 = A x^{3} + A + B x^{3} - B x^{2} + B x + C x (x^{2} + x) + D (x^{2} + x)$

Step 1.8.17.2

Apply the distributive property.

$x^{3} - 1 = A x^{3} + A + B x^{3} - B x^{2} + B x + C x \cdot x^{2} + C x \cdot x + D (x^{2} + x)$

Step 1.8.17.3

Apply the distributive property.

$x^{3} - 1 = A x^{3} + A + B x^{3} - B x^{2} + B x + C x \cdot x^{2} + C x \cdot x + D x^{2} + D x$

Step 1.8.18

Simplify each term.

Tap for more steps...

Step 1.8.18.1

Multiply $x$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 1.8.18.1.1

Move $x^{2}$ .

$x^{3} - 1 = A x^{3} + A + B x^{3} - B x^{2} + B x + C (x^{2} x) + C x \cdot x + D x^{2} + D x$

Step 1.8.18.1.2

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 1.8.18.1.2.1

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

$x^{3} - 1 = A x^{3} + A + B x^{3} - B x^{2} + B x + C (x^{2} x) + C x \cdot x + D x^{2} + D x$

Step 1.8.18.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{3} - 1 = A x^{3} + A + B x^{3} - B x^{2} + B x + C x^{2 + 1} + C x \cdot x + D x^{2} + D x$

Step 1.8.18.1.3

Add $2$ and $1$ .

$x^{3} - 1 = A x^{3} + A + B x^{3} - B x^{2} + B x + C x^{3} + C x \cdot x + D x^{2} + D x$

Step 1.8.18.2

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

Tap for more steps...

Step 1.8.18.2.1

Move $x$ .

$x^{3} - 1 = A x^{3} + A + B x^{3} - B x^{2} + B x + C x^{3} + C (x \cdot x) + D x^{2} + D x$

Step 1.8.18.2.2

Multiply $x$ by $x$ .

$x^{3} - 1 = A x^{3} + A + B x^{3} - B x^{2} + B x + C x^{3} + C x^{2} + D x^{2} + D x$

Step 1.9

Simplify the expression.

Tap for more steps...

Step 1.9.1

Move $B$ .

$x^{3} - 1 = A x^{3} + A + B x^{3} - x^{2} B + B x + C x^{3} + C x^{2} + D x^{2} + D x$

Step 1.9.2

Reorder $B$ and $x$ .

$x^{3} - 1 = A x^{3} + A + B x^{3} - x^{2} B + B x + C x^{3} + C x^{2} + D x^{2} + D x$

Step 1.9.3

Move $A$ .

$x^{3} - 1 = A x^{3} + B x^{3} - x^{2} B + B x + C x^{3} + C x^{2} + D x^{2} + D x + A$

Step 1.9.4

Move $B x$ .

$x^{3} - 1 = A x^{3} + B x^{3} - x^{2} B + C x^{3} + C x^{2} + D x^{2} + B x + D x + A$

Step 1.9.5

Move $- x^{2} B$ .

$x^{3} - 1 = A x^{3} + B x^{3} + C x^{3} - x^{2} B + C x^{2} + D x^{2} + B x + D x + A$

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 $x^{3}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = A + B + C$

Step 2.2

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.

$0 = - 1 B + C + D$

Step 2.3

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.

$0 = B + D$

Step 2.4

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 2.5

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

$1 = A + B + C$

$0 = - 1 B + C + D$

$0 = B + D$

$- 1 = A$

$1 = A + B + C$

$0 = - 1 B + C + D$

$0 = B + D$

$- 1 = A$

Step 3

Solve the system of equations.

Tap for more steps...

Step 3.1

Rewrite the equation as $A = - 1$ .

$A = - 1$

$1 = A + B + C$

$0 = - 1 B + C + D$

$0 = B + D$

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

$1 = (- 1) + B + C$

$A = - 1$

$0 = - 1 B + C + D$

$0 = B + D$

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

Remove parentheses.

$1 = - 1 + B + C$

$A = - 1$

$0 = - 1 B + C + D$

$0 = B + D$

$1 = - 1 + B + C$

$A = - 1$

$0 = - 1 B + C + D$

$0 = B + D$

$1 = - 1 + B + C$

$A = - 1$

$0 = - 1 B + C + D$

$0 = B + D$

Step 3.3

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

Tap for more steps...

Step 3.3.1

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

$- 1 + B + C = 1$

$A = - 1$

$0 = - 1 B + C + D$

$0 = B + D$

Step 3.3.2

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

Tap for more steps...

Step 3.3.2.1

Add $1$ to both sides of the equation.

$B + C = 1 + 1$

$A = - 1$

$0 = - 1 B + C + D$

$0 = B + D$

Step 3.3.2.2

Subtract $C$ from both sides of the equation.

$B = 1 + 1 - C$

$A = - 1$

$0 = - 1 B + C + D$

$0 = B + D$

Step 3.3.2.3

Add $1$ and $1$ .

$B = 2 - C$

$A = - 1$

$0 = - 1 B + C + D$

$0 = B + D$

$B = 2 - C$

$A = - 1$

$0 = - 1 B + C + D$

$0 = B + D$

$B = 2 - C$

$A = - 1$

$0 = - 1 B + C + D$

$0 = B + D$

Step 3.4

Replace all occurrences of $B$ with $2 - C$ in each equation.

Tap for more steps...

Step 3.4.1

Replace all occurrences of $B$ in $0 = - 1 B + C + D$ with $2 - C$ .

$0 = - 1 (2 - C) + C + D$

$B = 2 - C$

$A = - 1$

$0 = B + D$

Step 3.4.2

Simplify the right side.

Tap for more steps...

Step 3.4.2.1

Simplify $- 1 (2 - C) + C + D$ .

Tap for more steps...

Step 3.4.2.1.1

Simplify each term.

Tap for more steps...

Step 3.4.2.1.1.1

Apply the distributive property.

$0 = - 1 \cdot 2 - 1 (- C) + C + D$

$B = 2 - C$

$A = - 1$

$0 = B + D$

Step 3.4.2.1.1.2

Multiply $- 1$ by $2$ .

$0 = - 2 - 1 (- C) + C + D$

$B = 2 - C$

$A = - 1$

$0 = B + D$

Step 3.4.2.1.1.3

Multiply $- 1 (- C)$ .

Tap for more steps...

Step 3.4.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$0 = - 2 + 1 C + C + D$

$B = 2 - C$

$A = - 1$

$0 = B + D$

Step 3.4.2.1.1.3.2

Multiply $C$ by $1$ .

$0 = - 2 + C + C + D$

$B = 2 - C$

$A = - 1$

$0 = B + D$

$0 = - 2 + C + C + D$

$B = 2 - C$

$A = - 1$

$0 = B + D$

$0 = - 2 + C + C + D$

$B = 2 - C$

$A = - 1$

$0 = B + D$

Step 3.4.2.1.2

Add $C$ and $C$ .

$0 = - 2 + 2 C + D$

$B = 2 - C$

$A = - 1$

$0 = B + D$

$0 = - 2 + 2 C + D$

$B = 2 - C$

$A = - 1$

$0 = B + D$

$0 = - 2 + 2 C + D$

$B = 2 - C$

$A = - 1$

$0 = B + D$

Step 3.4.3

Replace all occurrences of $B$ in $0 = B + D$ with $2 - C$ .

$0 = (2 - C) + D$

$0 = - 2 + 2 C + D$

$B = 2 - C$

$A = - 1$

Step 3.4.4

Simplify the right side.

Tap for more steps...

Step 3.4.4.1

Remove parentheses.

$0 = 2 - C + D$

$0 = - 2 + 2 C + D$

$B = 2 - C$

$A = - 1$

$0 = 2 - C + D$

$0 = - 2 + 2 C + D$

$B = 2 - C$

$A = - 1$

$0 = 2 - C + D$

$0 = - 2 + 2 C + D$

$B = 2 - C$

$A = - 1$

Step 3.5

Reorder $2$ and $- C$ .

$B = - C + 2$

$0 = 2 - C + D$

$0 = - 2 + 2 C + D$

$A = - 1$

Step 3.6

Solve for $D$ in $0 = 2 - C + D$ .

Tap for more steps...

Step 3.6.1

Rewrite the equation as $2 - C + D = 0$ .

$2 - C + D = 0$

$B = - C + 2$

$0 = - 2 + 2 C + D$

$A = - 1$

Step 3.6.2

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

Tap for more steps...

Step 3.6.2.1

Subtract $2$ from both sides of the equation.

$- C + D = - 2$

$B = - C + 2$

$0 = - 2 + 2 C + D$

$A = - 1$

Step 3.6.2.2

Add $C$ to both sides of the equation.

$D = - 2 + C$

$B = - C + 2$

$0 = - 2 + 2 C + D$

$A = - 1$

$D = - 2 + C$

$B = - C + 2$

$0 = - 2 + 2 C + D$

$A = - 1$

$D = - 2 + C$

$B = - C + 2$

$0 = - 2 + 2 C + D$

$A = - 1$

Step 3.7

Replace all occurrences of $D$ with $- 2 + C$ in each equation.

Tap for more steps...

Step 3.7.1

Replace all occurrences of $D$ in $0 = - 2 + 2 C + D$ with $- 2 + C$ .

$0 = - 2 + 2 C - 2 + C$

$D = - 2 + C$

$B = - C + 2$

$A = - 1$

Step 3.7.2

Simplify $0 = - 2 + 2 C - 2 + C$ .

Tap for more steps...

Step 3.7.2.1

Simplify the left side.

Tap for more steps...

Step 3.7.2.1.1

Remove parentheses.

$0 = - 2 + 2 C - 2 + C$

$D = - 2 + C$

$B = - C + 2$

$A = - 1$

$0 = - 2 + 2 C - 2 + C$

$D = - 2 + C$

$B = - C + 2$

$A = - 1$

Step 3.7.2.2

Simplify the right side.

Tap for more steps...

Step 3.7.2.2.1

Simplify $- 2 + 2 C - 2 + C$ .

Tap for more steps...

Step 3.7.2.2.1.1

Subtract $2$ from $- 2$ .

$0 = 2 C - 4 + C$

$D = - 2 + C$

$B = - C + 2$

$A = - 1$

Step 3.7.2.2.1.2

Add $2 C$ and $C$ .

$0 = 3 C - 4$

$D = - 2 + C$

$B = - C + 2$

$A = - 1$

$0 = 3 C - 4$

$D = - 2 + C$

$B = - C + 2$

$A = - 1$

$0 = 3 C - 4$

$D = - 2 + C$

$B = - C + 2$

$A = - 1$

$0 = 3 C - 4$

$D = - 2 + C$

$B = - C + 2$

$A = - 1$

$0 = 3 C - 4$

$D = - 2 + C$

$B = - C + 2$

$A = - 1$

Step 3.8

Solve for $C$ in $0 = 3 C - 4$ .

Tap for more steps...

Step 3.8.1

Rewrite the equation as $3 C - 4 = 0$ .

$3 C - 4 = 0$

$D = - 2 + C$

$B = - C + 2$

$A = - 1$

Step 3.8.2

Add $4$ to both sides of the equation.

$3 C = 4$

$D = - 2 + C$

$B = - C + 2$

$A = - 1$

Step 3.8.3

Divide each term in $3 C = 4$ by $3$ and simplify.

Tap for more steps...

Step 3.8.3.1

Divide each term in $3 C = 4$ by $3$ .

$\frac{3 C}{3} = \frac{4}{3}$

$D = - 2 + C$

$B = - C + 2$

$A = - 1$

Step 3.8.3.2

Simplify the left side.

Tap for more steps...

Step 3.8.3.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.8.3.2.1.1

Cancel the common factor.

$\frac{3 C}{3} = \frac{4}{3}$

$D = - 2 + C$

$B = - C + 2$

$A = - 1$

Step 3.8.3.2.1.2

Divide $C$ by $1$ .

$C = \frac{4}{3}$

$D = - 2 + C$

$B = - C + 2$

$A = - 1$

$C = \frac{4}{3}$

$D = - 2 + C$

$B = - C + 2$

$A = - 1$

$C = \frac{4}{3}$

$D = - 2 + C$

$B = - C + 2$

$A = - 1$

$C = \frac{4}{3}$

$D = - 2 + C$

$B = - C + 2$

$A = - 1$

$C = \frac{4}{3}$

$D = - 2 + C$

$B = - C + 2$

$A = - 1$

Step 3.9

Replace all occurrences of $C$ with $\frac{4}{3}$ in each equation.

Tap for more steps...

Step 3.9.1

Replace all occurrences of $C$ in $D = - 2 + C$ with $\frac{4}{3}$ .

$D = - 2 + \frac{4}{3}$

$C = \frac{4}{3}$

$B = - C + 2$

$A = - 1$

Step 3.9.2

Simplify $D = - 2 + \frac{4}{3}$ .

Tap for more steps...

Step 3.9.2.1

Simplify the left side.

Tap for more steps...

Step 3.9.2.1.1

Remove parentheses.

$D = - 2 + \frac{4}{3}$

$C = \frac{4}{3}$

$B = - C + 2$

$A = - 1$

$D = - 2 + \frac{4}{3}$

$C = \frac{4}{3}$

$B = - C + 2$

$A = - 1$

Step 3.9.2.2

Simplify the right side.

Tap for more steps...

Step 3.9.2.2.1

Simplify $- 2 + \frac{4}{3}$ .

Tap for more steps...

Step 3.9.2.2.1.1

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$D = - 2 \cdot \frac{3}{3} + \frac{4}{3}$

$C = \frac{4}{3}$

$B = - C + 2$

$A = - 1$

Step 3.9.2.2.1.2

Combine $- 2$ and $\frac{3}{3}$ .

$D = \frac{- 2 \cdot 3}{3} + \frac{4}{3}$

$C = \frac{4}{3}$

$B = - C + 2$

$A = - 1$

Step 3.9.2.2.1.3

Combine the numerators over the common denominator.

$D = \frac{- 2 \cdot 3 + 4}{3}$

$C = \frac{4}{3}$

$B = - C + 2$

$A = - 1$

Step 3.9.2.2.1.4

Simplify the numerator.

Tap for more steps...

Step 3.9.2.2.1.4.1

Multiply $- 2$ by $3$ .

$D = \frac{- 6 + 4}{3}$

$C = \frac{4}{3}$

$B = - C + 2$

$A = - 1$

Step 3.9.2.2.1.4.2

Add $- 6$ and $4$ .

$D = \frac{- 2}{3}$

$C = \frac{4}{3}$

$B = - C + 2$

$A = - 1$

$D = \frac{- 2}{3}$

$C = \frac{4}{3}$

$B = - C + 2$

$A = - 1$

Step 3.9.2.2.1.5

Move the negative in front of the fraction.

$D = - \frac{2}{3}$

$C = \frac{4}{3}$

$B = - C + 2$

$A = - 1$

$D = - \frac{2}{3}$

$C = \frac{4}{3}$

$B = - C + 2$

$A = - 1$

$D = - \frac{2}{3}$

$C = \frac{4}{3}$

$B = - C + 2$

$A = - 1$

$D = - \frac{2}{3}$

$C = \frac{4}{3}$

$B = - C + 2$

$A = - 1$

Step 3.9.3

Replace all occurrences of $C$ in $B = - C + 2$ with $\frac{4}{3}$ .

$B = - (\frac{4}{3}) + 2$

$D = - \frac{2}{3}$

$C = \frac{4}{3}$

$A = - 1$

Step 3.9.4

Simplify the right side.

Tap for more steps...

Step 3.9.4.1

Simplify $- (\frac{4}{3}) + 2$ .

Tap for more steps...

Step 3.9.4.1.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$B = - \frac{4}{3} + 2 \cdot \frac{3}{3}$

$D = - \frac{2}{3}$

$C = \frac{4}{3}$

$A = - 1$

Step 3.9.4.1.2

Combine $2$ and $\frac{3}{3}$ .

$B = - \frac{4}{3} + \frac{2 \cdot 3}{3}$

$D = - \frac{2}{3}$

$C = \frac{4}{3}$

$A = - 1$

Step 3.9.4.1.3

Combine the numerators over the common denominator.

$B = \frac{- 4 + 2 \cdot 3}{3}$

$D = - \frac{2}{3}$

$C = \frac{4}{3}$

$A = - 1$

Step 3.9.4.1.4

Simplify the numerator.

Tap for more steps...

Step 3.9.4.1.4.1

Multiply $2$ by $3$ .

$B = \frac{- 4 + 6}{3}$

$D = - \frac{2}{3}$

$C = \frac{4}{3}$

$A = - 1$

Step 3.9.4.1.4.2

Add $- 4$ and $6$ .

$B = \frac{2}{3}$

$D = - \frac{2}{3}$

$C = \frac{4}{3}$

$A = - 1$

$B = \frac{2}{3}$

$D = - \frac{2}{3}$

$C = \frac{4}{3}$

$A = - 1$

$B = \frac{2}{3}$

$D = - \frac{2}{3}$

$C = \frac{4}{3}$

$A = - 1$

$B = \frac{2}{3}$

$D = - \frac{2}{3}$

$C = \frac{4}{3}$

$A = - 1$

$B = \frac{2}{3}$

$D = - \frac{2}{3}$

$C = \frac{4}{3}$

$A = - 1$

Step 3.10

List all of the solutions.

$B = \frac{2}{3}, D = - \frac{2}{3}, C = \frac{4}{3}, A = - 1$

Step 4

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

$- \frac{1}{x} + \frac{\frac{2}{3}}{x + 1} + \frac{\frac{4}{3 x} - \frac{2}{3}}{x^{2} - x + 1}$

Step 5

Combine $\frac{4}{3}$ and $x$ .

$\frac{- 1}{x} + \frac{\frac{2}{3}}{x + 1} + \frac{\frac{4 x}{3} - \frac{2}{3}}{x^{2} - x + 1}$