Precalculus Examples

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

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

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

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Step 1.1.1

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

$\frac{x^{2} + 2 x + 3}{x \cdot x^{2} + x}$

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

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

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.

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

Step 1.3

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

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

Step 1.4

Cancel the common factor of $x$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

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

Step 1.5

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

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Step 1.5.1

Cancel the common factor.

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

Step 1.5.2

Divide $x^{2} + 2 x + 3$ by $1$ .

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

Step 1.6

Simplify each term.

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Step 1.6.1

Cancel the common factor of $x$ .

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Step 1.6.1.1

Cancel the common factor.

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

Step 1.6.1.2

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

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

Step 1.6.2

Apply the distributive property.

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

Step 1.6.3

Multiply $A$ by $1$ .

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

Step 1.6.4

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

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Step 1.6.4.1

Cancel the common factor.

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

Step 1.6.4.2

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

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

Step 1.6.5

Apply the distributive property.

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

Step 1.6.6

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

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Step 1.6.6.1

Move $x$ .

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

Step 1.6.6.2

Multiply $x$ by $x$ .

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

Step 1.7

Move $A$ .

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

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.

$2 = C$

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.

$3 = A$

Step 2.4

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

$1 = A + B$

$2 = C$

$3 = A$

$1 = A + B$

$2 = C$

$3 = A$

Step 3

Solve the system of equations.

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Step 3.1

Rewrite the equation as $C = 2$ .

$C = 2$

$1 = A + B$

$3 = A$

Step 3.2

Rewrite the equation as $A = 3$ .

$A = 3$

$C = 2$

$1 = A + B$

Step 3.3

Replace all occurrences of $A$ with $3$ in each equation.

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Step 3.3.1

Replace all occurrences of $A$ in $1 = A + B$ with $3$ .

$1 = (3) + B$

$A = 3$

$C = 2$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Remove parentheses.

$1 = 3 + B$

$A = 3$

$C = 2$

$1 = 3 + B$

$A = 3$

$C = 2$

$1 = 3 + B$

$A = 3$

$C = 2$

Step 3.4

Solve for $B$ in $1 = 3 + B$ .

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Step 3.4.1

Rewrite the equation as $3 + B = 1$ .

$3 + B = 1$

$A = 3$

$C = 2$

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

Subtract $3$ from both sides of the equation.

$B = 1 - 3$

$A = 3$

$C = 2$

Step 3.4.2.2

Subtract $3$ from $1$ .

$B = - 2$

$A = 3$

$C = 2$

$B = - 2$

$A = 3$

$C = 2$

$B = - 2$

$A = 3$

$C = 2$

Step 3.5

Solve the system of equations.

$B = - 2 A = 3 C = 2$

Step 3.6

List all of the solutions.

$B = - 2, A = 3, C = 2$

Step 4

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

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