Precalculus Examples

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

Step 1

Divide using long polynomial division.

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Step 1.1

Expand $x (x^{2} + 1)$ .

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Reorder $x$ and $1$ .

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

Add $1$ and $2$ .

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

Step 1.1.6

Multiply $x$ by $1$ .

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

Step 1.2

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$0 x$

$1$

Step 1.3

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x^{3}$ .

$x$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$0 x$

$1$

Step 1.4

Multiply the new quotient term by the divisor.

$x$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$0 x$

$1$

$x^{4}$

$0$

$x^{2}$

$0$

Step 1.5

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + 0 + x^{2} + 0$

$x$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$0 x$

$1$

$x^{4}$

$0$

$x^{2}$

$0$

Step 1.6

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$0 x$

$1$

$x^{4}$

$0$

$x^{2}$

$0$

$2 x^{2}$

$0$

Step 1.7

Pull the next terms from the original dividend down into the current dividend.

$x$

$x^{3}$

$0 x^{2}$

$x$

$0$

$x^{4}$

$0 x^{3}$

$3 x^{2}$

$0 x$

$1$

$x^{4}$

$0$

$x^{2}$

$0$

$2 x^{2}$

$0$

$1$

Step 1.8

The final answer is the quotient plus the remainder over the divisor.

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

Step 2

Decompose the fraction and multiply through by the common denominator.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.2

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 2.3

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

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

Step 2.4

Cancel the common factor of $x$ .

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Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

Rewrite the expression.

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

Step 2.5

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

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Step 2.5.1

Cancel the common factor.

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

Step 2.5.2

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

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

Step 2.6

Simplify each term.

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Step 2.6.1

Cancel the common factor of $x$ .

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Step 2.6.1.1

Cancel the common factor.

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

Step 2.6.1.2

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

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

Step 2.6.2

Apply the distributive property.

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

Step 2.6.3

Multiply $A$ by $1$ .

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

Step 2.6.4

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

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

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

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

Step 2.6.5

Apply the distributive property.

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

Step 2.6.6

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

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Step 2.6.6.1

Move $x$ .

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

Step 2.6.6.2

Multiply $x$ by $x$ .

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

Step 2.7

Move $A$ .

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

Step 3

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

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Step 3.1

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.

$2 = A + B$

Step 3.2

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.

$0 = C$

Step 3.3

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 3.4

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

$2 = A + B$

$0 = C$

$1 = A$

$2 = A + B$

$0 = C$

$1 = A$

Step 4

Solve the system of equations.

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Step 4.1

Rewrite the equation as $C = 0$ .

$C = 0$

$2 = A + B$

$1 = A$

Step 4.2

Rewrite the equation as $A = 1$ .

$A = 1$

$C = 0$

$2 = A + B$

Step 4.3

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

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Step 4.3.1

Replace all occurrences of $A$ in $2 = A + B$ with $1$ .

$2 = (1) + B$

$A = 1$

$C = 0$

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Remove parentheses.

$2 = 1 + B$

$A = 1$

$C = 0$

$2 = 1 + B$

$A = 1$

$C = 0$

$2 = 1 + B$

$A = 1$

$C = 0$

Step 4.4

Solve for $B$ in $2 = 1 + B$ .

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Step 4.4.1

Rewrite the equation as $1 + B = 2$ .

$1 + B = 2$

$A = 1$

$C = 0$

Step 4.4.2

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

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Step 4.4.2.1

Subtract $1$ from both sides of the equation.

$B = 2 - 1$

$A = 1$

$C = 0$

Step 4.4.2.2

Subtract $1$ from $2$ .

$B = 1$

$A = 1$

$C = 0$

$B = 1$

$A = 1$

$C = 0$

$B = 1$

$A = 1$

$C = 0$

Step 4.5

Solve the system of equations.

$B = 1 A = 1 C = 0$

Step 4.6

List all of the solutions.

$B = 1, A = 1, C = 0$

Step 5

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

$x + \frac{1}{x} + \frac{1 x + 0}{x^{2} + 1}$

Step 6

Simplify.

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Step 6.1

Remove parentheses.

$x + \frac{1}{x} + \frac{(1) \cdot x + 0}{x^{2} + 1}$

Step 6.2

Add $(1) \cdot x$ and $0$ .

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

Step 6.3

Multiply $x$ by $1$ .

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