Calculus Examples

$\frac{x^{3}}{x^{3} - 3 x^{2} + 9 x - 27}$

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

Factor out the greatest common factor from each group.

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Step 1.1.1.1

Group the first two terms and the last two terms.

$\frac{x^{3}}{(x^{3} - 3 x^{2}) + 9 x - 27}$

Step 1.1.1.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x^{3}}{x^{2} (x - 3) + 9 (x - 3)}$

Step 1.1.2

Factor the polynomial by factoring out the greatest common factor, $x - 3$ .

$\frac{x^{3}}{(x - 3) (x^{2} + 9)}$

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 $A$ .

$\frac{A}{x - 3}$

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.

$\frac{A}{x - 3} + \frac{B x + C}{x^{2} + 9}$

Step 1.4

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(x - 3) (x^{2} + 9)$ .

$\frac{x^{3} (x - 3) (x^{2} + 9)}{(x - 3) (x^{2} + 9)} = \frac{(A) (x - 3) (x^{2} + 9)}{x - 3} + \frac{(B x + C) (x - 3) (x^{2} + 9)}{x^{2} + 9}$

Step 1.5

Cancel the common factor of $x - 3$ .

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Step 1.5.1

Cancel the common factor.

$\frac{x^{3} (x - 3) (x^{2} + 9)}{(x - 3) (x^{2} + 9)} = \frac{(A) (x - 3) (x^{2} + 9)}{x - 3} + \frac{(B x + C) (x - 3) (x^{2} + 9)}{x^{2} + 9}$

Step 1.5.2

Rewrite the expression.

$\frac{x^{3} (x^{2} + 9)}{x^{2} + 9} = \frac{(A) (x - 3) (x^{2} + 9)}{x - 3} + \frac{(B x + C) (x - 3) (x^{2} + 9)}{x^{2} + 9}$

Step 1.6

Cancel the common factor of $x^{2} + 9$ .

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Step 1.6.1

Cancel the common factor.

$\frac{x^{3} (x^{2} + 9)}{x^{2} + 9} = \frac{(A) (x - 3) (x^{2} + 9)}{x - 3} + \frac{(B x + C) (x - 3) (x^{2} + 9)}{x^{2} + 9}$

Step 1.6.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{(A) (x - 3) (x^{2} + 9)}{x - 3} + \frac{(B x + C) (x - 3) (x^{2} + 9)}{x^{2} + 9}$

Step 1.7

Simplify each term.

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Step 1.7.1

Cancel the common factor of $x - 3$ .

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Step 1.7.1.1

Cancel the common factor.

$x^{3} = \frac{A (x - 3) (x^{2} + 9)}{x - 3} + \frac{(B x + C) (x - 3) (x^{2} + 9)}{x^{2} + 9}$

Step 1.7.1.2

Divide $(A) (x^{2} + 9)$ by $1$ .

$x^{3} = (A) (x^{2} + 9) + \frac{(B x + C) (x - 3) (x^{2} + 9)}{x^{2} + 9}$

Step 1.7.2

Apply the distributive property.

$x^{3} = A x^{2} + A \cdot 9 + \frac{(B x + C) (x - 3) (x^{2} + 9)}{x^{2} + 9}$

Step 1.7.3

Move $9$ to the left of $A$ .

$x^{3} = A x^{2} + 9 \cdot A + \frac{(B x + C) (x - 3) (x^{2} + 9)}{x^{2} + 9}$

Step 1.7.4

Cancel the common factor of $x^{2} + 9$ .

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Step 1.7.4.1

Cancel the common factor.

$x^{3} = A x^{2} + 9 A + \frac{(B x + C) (x - 3) (x^{2} + 9)}{x^{2} + 9}$

Step 1.7.4.2

Divide $(B x + C) (x - 3)$ by $1$ .

$x^{3} = A x^{2} + 9 A + (B x + C) (x - 3)$

Step 1.7.5

Expand $(B x + C) (x - 3)$ using the FOIL Method.

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Step 1.7.5.1

Apply the distributive property.

$x^{3} = A x^{2} + 9 A + B x (x - 3) + C (x - 3)$

Step 1.7.5.2

Apply the distributive property.

$x^{3} = A x^{2} + 9 A + B x \cdot x + B x \cdot - 3 + C (x - 3)$

Step 1.7.5.3

Apply the distributive property.

$x^{3} = A x^{2} + 9 A + B x \cdot x + B x \cdot - 3 + C x + C \cdot - 3$

Step 1.7.6

Simplify each term.

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Step 1.7.6.1

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

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Step 1.7.6.1.1

Move $x$ .

$x^{3} = A x^{2} + 9 A + B (x \cdot x) + B x \cdot - 3 + C x + C \cdot - 3$

Step 1.7.6.1.2

Multiply $x$ by $x$ .

$x^{3} = A x^{2} + 9 A + B x^{2} + B x \cdot - 3 + C x + C \cdot - 3$

Step 1.7.6.2

Move $- 3$ to the left of $B x$ .

$x^{3} = A x^{2} + 9 A + B x^{2} - 3 \cdot (B x) + C x + C \cdot - 3$

Step 1.7.6.3

Move $- 3$ to the left of $C$ .

$x^{3} = A x^{2} + 9 A + B x^{2} - 3 B x + C x - 3 C$

Step 1.8

Simplify the expression.

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Step 1.8.1

Move $B$ .

$x^{3} = A x^{2} + 9 A + B x^{2} - 3 x B + C x - 3 C$

Step 1.8.2

Move $9 A$ .

$x^{3} = A x^{2} + B x^{2} - 3 x B + C x + 9 A - 3 C$

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.

$0 = A + B$

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.

$0 = - 3 B + C$

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.

$0 = 9 A - 3 C$

Step 2.4

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

$0 = A + B$

$0 = - 3 B + C$

$0 = 9 A - 3 C$

$0 = A + B$

$0 = - 3 B + C$

$0 = 9 A - 3 C$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $A$ in $0 = A + B$ .

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Step 3.1.1

Rewrite the equation as $A + B = 0$ .

$A + B = 0$

$0 = - 3 B + C$

$0 = 9 A - 3 C$

Step 3.1.2

Subtract $B$ from both sides of the equation.

$A = - B$

$0 = - 3 B + C$

$0 = 9 A - 3 C$

$A = - B$

$0 = - 3 B + C$

$0 = 9 A - 3 C$

Step 3.2

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

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Step 3.2.1

Replace all occurrences of $A$ in $0 = 9 A - 3 C$ with $- B$ .

$0 = 9 (- B) - 3 C$

$A = - B$

$0 = - 3 B + C$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- 1$ by $9$ .

$0 = - 9 B - 3 C$

$A = - B$

$0 = - 3 B + C$

$0 = - 9 B - 3 C$

$A = - B$

$0 = - 3 B + C$

$0 = - 9 B - 3 C$

$A = - B$

$0 = - 3 B + C$

Step 3.3

Solve for $C$ in $0 = - 3 B + C$ .

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Step 3.3.1

Rewrite the equation as $- 3 B + C = 0$ .

$- 3 B + C = 0$

$0 = - 9 B - 3 C$

$A = - B$

Step 3.3.2

Add $3 B$ to both sides of the equation.

$C = 3 B$

$0 = - 9 B - 3 C$

$A = - B$

$C = 3 B$

$0 = - 9 B - 3 C$

$A = - B$

Step 3.4

Replace all occurrences of $C$ with $3 B$ in each equation.

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Step 3.4.1

Replace all occurrences of $C$ in $0 = - 9 B - 3 C$ with $3 B$ .

$0 = - 9 B - 3 (3 B)$

$C = 3 B$

$A = - B$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- 9 B - 3 (3 B)$ .

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Step 3.4.2.1.1

Multiply $3$ by $- 3$ .

$0 = - 9 B - 9 B$

$C = 3 B$

$A = - B$

Step 3.4.2.1.2

Subtract $9 B$ from $- 9 B$ .

$0 = - 18 B$

$C = 3 B$

$A = - B$

$0 = - 18 B$

$C = 3 B$

$A = - B$

$0 = - 18 B$

$C = 3 B$

$A = - B$

$0 = - 18 B$

$C = 3 B$

$A = - B$

Step 3.5

Solve for $B$ in $0 = - 18 B$ .

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Step 3.5.1

Rewrite the equation as $- 18 B = 0$ .

$- 18 B = 0$

$C = 3 B$

$A = - B$

Step 3.5.2

Divide each term in $- 18 B = 0$ by $- 18$ and simplify.

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Step 3.5.2.1

Divide each term in $- 18 B = 0$ by $- 18$ .

$\frac{- 18 B}{- 18} = \frac{0}{- 18}$

$C = 3 B$

$A = - B$

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Cancel the common factor of $- 18$ .

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Step 3.5.2.2.1.1

Cancel the common factor.

$\frac{- 18 B}{- 18} = \frac{0}{- 18}$

$C = 3 B$

$A = - B$

Step 3.5.2.2.1.2

Divide $B$ by $1$ .

$B = \frac{0}{- 18}$

$C = 3 B$

$A = - B$

$B = \frac{0}{- 18}$

$C = 3 B$

$A = - B$

$B = \frac{0}{- 18}$

$C = 3 B$

$A = - B$

Step 3.5.2.3

Simplify the right side.

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Step 3.5.2.3.1

Divide $0$ by $- 18$ .

$B = 0$

$C = 3 B$

$A = - B$

$B = 0$

$C = 3 B$

$A = - B$

$B = 0$

$C = 3 B$

$A = - B$

$B = 0$

$C = 3 B$

$A = - B$

Step 3.6

Replace all occurrences of $B$ with $0$ in each equation.

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Step 3.6.1

Replace all occurrences of $B$ in $C = 3 B$ with $0$ .

$C = 3 (0)$

$B = 0$

$A = - B$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Multiply $3$ by $0$ .

$C = 0$

$B = 0$

$A = - B$

$C = 0$

$B = 0$

$A = - B$

Step 3.6.3

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

$A = - (0)$

$C = 0$

$B = 0$

Step 3.6.4

Simplify the right side.

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Step 3.6.4.1

Multiply $- 1$ by $0$ .

$A = 0$

$C = 0$

$B = 0$

$A = 0$

$C = 0$

$B = 0$

$A = 0$

$C = 0$

$B = 0$

Step 3.7

List all of the solutions.

$A = 0, C = 0, B = 0$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x - 3} + \frac{B x + C}{x^{2} + 9}$ with the values found for $A$ , $B$ , and $C$ .

$\frac{0}{x - 3} + \frac{0 x + 0}{x^{2} + 9}$

Step 5

Simplify.

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Step 5.1

Add $(0) \cdot x$ and $0$ .

$\frac{0}{x - 3} + \frac{(0) \cdot x}{x^{2} + 9}$

Step 5.2

Multiply $0$ by $x$ .

$\frac{0}{x - 3} + \frac{0}{x^{2} + 9}$