Precalculus Examples

$\frac{5 x^{2} + 14 x + 18}{(x + 6) (x^{2} + 2)}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

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 $A$ .

$\frac{A}{x + 6}$

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.

$\frac{A}{x + 6} + \frac{B x + C}{x^{2} + 2}$

Step 1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(x + 6) (x^{2} + 2)$ .

$\frac{(5 x^{2} + 14 x + 18) (x + 6) (x^{2} + 2)}{(x + 6) (x^{2} + 2)} = \frac{(A) (x + 6) (x^{2} + 2)}{x + 6} + \frac{(B x + C) (x + 6) (x^{2} + 2)}{x^{2} + 2}$

Step 1.4

Cancel the common factor of $x + 6$ .

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Step 1.4.1

Cancel the common factor.

$\frac{(5 x^{2} + 14 x + 18) (x + 6) (x^{2} + 2)}{(x + 6) (x^{2} + 2)} = \frac{(A) (x + 6) (x^{2} + 2)}{x + 6} + \frac{(B x + C) (x + 6) (x^{2} + 2)}{x^{2} + 2}$

Step 1.4.2

Rewrite the expression.

$\frac{(5 x^{2} + 14 x + 18) (x^{2} + 2)}{x^{2} + 2} = \frac{(A) (x + 6) (x^{2} + 2)}{x + 6} + \frac{(B x + C) (x + 6) (x^{2} + 2)}{x^{2} + 2}$

Step 1.5

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

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Step 1.5.1

Cancel the common factor.

$\frac{(5 x^{2} + 14 x + 18) (x^{2} + 2)}{x^{2} + 2} = \frac{(A) (x + 6) (x^{2} + 2)}{x + 6} + \frac{(B x + C) (x + 6) (x^{2} + 2)}{x^{2} + 2}$

Step 1.5.2

Divide $5 x^{2} + 14 x + 18$ by $1$ .

$5 x^{2} + 14 x + 18 = \frac{(A) (x + 6) (x^{2} + 2)}{x + 6} + \frac{(B x + C) (x + 6) (x^{2} + 2)}{x^{2} + 2}$

Step 1.6

Simplify each term.

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Step 1.6.1

Cancel the common factor of $x + 6$ .

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Step 1.6.1.1

Cancel the common factor.

$5 x^{2} + 14 x + 18 = \frac{A (x + 6) (x^{2} + 2)}{x + 6} + \frac{(B x + C) (x + 6) (x^{2} + 2)}{x^{2} + 2}$

Step 1.6.1.2

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

$5 x^{2} + 14 x + 18 = (A) (x^{2} + 2) + \frac{(B x + C) (x + 6) (x^{2} + 2)}{x^{2} + 2}$

Step 1.6.2

Apply the distributive property.

$5 x^{2} + 14 x + 18 = A x^{2} + A \cdot 2 + \frac{(B x + C) (x + 6) (x^{2} + 2)}{x^{2} + 2}$

Step 1.6.3

Move $2$ to the left of $A$ .

$5 x^{2} + 14 x + 18 = A x^{2} + 2 \cdot A + \frac{(B x + C) (x + 6) (x^{2} + 2)}{x^{2} + 2}$

Step 1.6.4

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

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Step 1.6.4.1

Cancel the common factor.

$5 x^{2} + 14 x + 18 = A x^{2} + 2 A + \frac{(B x + C) (x + 6) (x^{2} + 2)}{x^{2} + 2}$

Step 1.6.4.2

Divide $(B x + C) (x + 6)$ by $1$ .

$5 x^{2} + 14 x + 18 = A x^{2} + 2 A + (B x + C) (x + 6)$

Step 1.6.5

Expand $(B x + C) (x + 6)$ using the FOIL Method.

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Step 1.6.5.1

Apply the distributive property.

$5 x^{2} + 14 x + 18 = A x^{2} + 2 A + B x (x + 6) + C (x + 6)$

Step 1.6.5.2

Apply the distributive property.

$5 x^{2} + 14 x + 18 = A x^{2} + 2 A + B x \cdot x + B x \cdot 6 + C (x + 6)$

Step 1.6.5.3

Apply the distributive property.

$5 x^{2} + 14 x + 18 = A x^{2} + 2 A + B x \cdot x + B x \cdot 6 + C x + C \cdot 6$

Step 1.6.6

Simplify each term.

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Step 1.6.6.1

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

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Step 1.6.6.1.1

Move $x$ .

$5 x^{2} + 14 x + 18 = A x^{2} + 2 A + B (x \cdot x) + B x \cdot 6 + C x + C \cdot 6$

Step 1.6.6.1.2

Multiply $x$ by $x$ .

$5 x^{2} + 14 x + 18 = A x^{2} + 2 A + B x^{2} + B x \cdot 6 + C x + C \cdot 6$

Step 1.6.6.2

Move $6$ to the left of $B x$ .

$5 x^{2} + 14 x + 18 = A x^{2} + 2 A + B x^{2} + 6 \cdot (B x) + C x + C \cdot 6$

Step 1.6.6.3

Move $6$ to the left of $C$ .

$5 x^{2} + 14 x + 18 = A x^{2} + 2 A + B x^{2} + 6 B x + C x + 6 C$

Step 1.7

Simplify the expression.

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Step 1.7.1

Move $B$ .

$5 x^{2} + 14 x + 18 = A x^{2} + 2 A + B x^{2} + 6 x B + C x + 6 C$

Step 1.7.2

Move $2 A$ .

$5 x^{2} + 14 x + 18 = A x^{2} + B x^{2} + 6 x B + C x + 2 A + 6 C$

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.

$5 = A + B$

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.

$14 = 6 B + C$

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.

$18 = 2 A + 6 C$

Step 2.4

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

$5 = A + B$

$14 = 6 B + C$

$18 = 2 A + 6 C$

$5 = A + B$

$14 = 6 B + C$

$18 = 2 A + 6 C$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $A$ in $5 = A + B$ .

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Step 3.1.1

Rewrite the equation as $A + B = 5$ .

$A + B = 5$

$14 = 6 B + C$

$18 = 2 A + 6 C$

Step 3.1.2

Subtract $B$ from both sides of the equation.

$A = 5 - B$

$14 = 6 B + C$

$18 = 2 A + 6 C$

$A = 5 - B$

$14 = 6 B + C$

$18 = 2 A + 6 C$

Step 3.2

Replace all occurrences of $A$ with $5 - B$ in each equation.

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Step 3.2.1

Replace all occurrences of $A$ in $18 = 2 A + 6 C$ with $5 - B$ .

$18 = 2 (5 - B) + 6 C$

$A = 5 - B$

$14 = 6 B + C$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Apply the distributive property.

$18 = 2 \cdot 5 + 2 (- B) + 6 C$

$A = 5 - B$

$14 = 6 B + C$

Step 3.2.2.1.2

Multiply $2$ by $5$ .

$18 = 10 + 2 (- B) + 6 C$

$A = 5 - B$

$14 = 6 B + C$

Step 3.2.2.1.3

Multiply $- 1$ by $2$ .

$18 = 10 - 2 B + 6 C$

$A = 5 - B$

$14 = 6 B + C$

$18 = 10 - 2 B + 6 C$

$A = 5 - B$

$14 = 6 B + C$

$18 = 10 - 2 B + 6 C$

$A = 5 - B$

$14 = 6 B + C$

$18 = 10 - 2 B + 6 C$

$A = 5 - B$

$14 = 6 B + C$

Step 3.3

Reorder $5$ and $- B$ .

$A = - B + 5$

$18 = 10 - 2 B + 6 C$

$14 = 6 B + C$

Step 3.4

Solve for $C$ in $14 = 6 B + C$ .

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Step 3.4.1

Rewrite the equation as $6 B + C = 14$ .

$6 B + C = 14$

$A = - B + 5$

$18 = 10 - 2 B + 6 C$

Step 3.4.2

Subtract $6 B$ from both sides of the equation.

$C = 14 - 6 B$

$A = - B + 5$

$18 = 10 - 2 B + 6 C$

$C = 14 - 6 B$

$A = - B + 5$

$18 = 10 - 2 B + 6 C$

Step 3.5

Replace all occurrences of $C$ with $14 - 6 B$ in each equation.

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Step 3.5.1

Replace all occurrences of $C$ in $18 = 10 - 2 B + 6 C$ with $14 - 6 B$ .

$18 = 10 - 2 B + 6 (14 - 6 B)$

$C = 14 - 6 B$

$A = - B + 5$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify $10 - 2 B + 6 (14 - 6 B)$ .

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Step 3.5.2.1.1

Simplify each term.

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Step 3.5.2.1.1.1

Apply the distributive property.

$18 = 10 - 2 B + 6 \cdot 14 + 6 (- 6 B)$

$C = 14 - 6 B$

$A = - B + 5$

Step 3.5.2.1.1.2

Multiply $6$ by $14$ .

$18 = 10 - 2 B + 84 + 6 (- 6 B)$

$C = 14 - 6 B$

$A = - B + 5$

Step 3.5.2.1.1.3

Multiply $- 6$ by $6$ .

$18 = 10 - 2 B + 84 - 36 B$

$C = 14 - 6 B$

$A = - B + 5$

$18 = 10 - 2 B + 84 - 36 B$

$C = 14 - 6 B$

$A = - B + 5$

Step 3.5.2.1.2

Simplify by adding terms.

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Step 3.5.2.1.2.1

Add $10$ and $84$ .

$18 = - 2 B + 94 - 36 B$

$C = 14 - 6 B$

$A = - B + 5$

Step 3.5.2.1.2.2

Subtract $36 B$ from $- 2 B$ .

$18 = - 38 B + 94$

$C = 14 - 6 B$

$A = - B + 5$

$18 = - 38 B + 94$

$C = 14 - 6 B$

$A = - B + 5$

$18 = - 38 B + 94$

$C = 14 - 6 B$

$A = - B + 5$

$18 = - 38 B + 94$

$C = 14 - 6 B$

$A = - B + 5$

$18 = - 38 B + 94$

$C = 14 - 6 B$

$A = - B + 5$

Step 3.6

Solve for $B$ in $18 = - 38 B + 94$ .

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Step 3.6.1

Rewrite the equation as $- 38 B + 94 = 18$ .

$- 38 B + 94 = 18$

$C = 14 - 6 B$

$A = - B + 5$

Step 3.6.2

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

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Step 3.6.2.1

Subtract $94$ from both sides of the equation.

$- 38 B = 18 - 94$

$C = 14 - 6 B$

$A = - B + 5$

Step 3.6.2.2

Subtract $94$ from $18$ .

$- 38 B = - 76$

$C = 14 - 6 B$

$A = - B + 5$

$- 38 B = - 76$

$C = 14 - 6 B$

$A = - B + 5$

Step 3.6.3

Divide each term in $- 38 B = - 76$ by $- 38$ and simplify.

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Step 3.6.3.1

Divide each term in $- 38 B = - 76$ by $- 38$ .

$\frac{- 38 B}{- 38} = \frac{- 76}{- 38}$

$C = 14 - 6 B$

$A = - B + 5$

Step 3.6.3.2

Simplify the left side.

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Step 3.6.3.2.1

Cancel the common factor of $- 38$ .

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Step 3.6.3.2.1.1

Cancel the common factor.

$\frac{- 38 B}{- 38} = \frac{- 76}{- 38}$

$C = 14 - 6 B$

$A = - B + 5$

Step 3.6.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{- 76}{- 38}$

$C = 14 - 6 B$

$A = - B + 5$

$B = \frac{- 76}{- 38}$

$C = 14 - 6 B$

$A = - B + 5$

$B = \frac{- 76}{- 38}$

$C = 14 - 6 B$

$A = - B + 5$

Step 3.6.3.3

Simplify the right side.

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Step 3.6.3.3.1

Divide $- 76$ by $- 38$ .

$B = 2$

$C = 14 - 6 B$

$A = - B + 5$

$B = 2$

$C = 14 - 6 B$

$A = - B + 5$

$B = 2$

$C = 14 - 6 B$

$A = - B + 5$

$B = 2$

$C = 14 - 6 B$

$A = - B + 5$

Step 3.7

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

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Step 3.7.1

Replace all occurrences of $B$ in $C = 14 - 6 B$ with $2$ .

$C = 14 - 6 \cdot 2$

$B = 2$

$A = - B + 5$

Step 3.7.2

Simplify the right side.

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Step 3.7.2.1

Simplify $14 - 6 \cdot 2$ .

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Step 3.7.2.1.1

Multiply $- 6$ by $2$ .

$C = 14 - 12$

$B = 2$

$A = - B + 5$

Step 3.7.2.1.2

Subtract $12$ from $14$ .

$C = 2$

$B = 2$

$A = - B + 5$

$C = 2$

$B = 2$

$A = - B + 5$

$C = 2$

$B = 2$

$A = - B + 5$

Step 3.7.3

Replace all occurrences of $B$ in $A = - B + 5$ with $2$ .

$A = - (2) + 5$

$C = 2$

$B = 2$

Step 3.7.4

Simplify the right side.

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Step 3.7.4.1

Simplify $- (2) + 5$ .

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Step 3.7.4.1.1

Multiply $- 1$ by $2$ .

$A = - 2 + 5$

$C = 2$

$B = 2$

Step 3.7.4.1.2

Add $- 2$ and $5$ .

$A = 3$

$C = 2$

$B = 2$

$A = 3$

$C = 2$

$B = 2$

$A = 3$

$C = 2$

$B = 2$

$A = 3$

$C = 2$

$B = 2$

Step 3.8

List all of the solutions.

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

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x + 6} + \frac{B x + C}{x^{2} + 2}$ with the values found for $A$ , $B$ , and $C$ .

$\frac{3}{x + 6} + \frac{2 x + 2}{x^{2} + 2}$

Step 5

Multiply $2$ by $x$ .

$\frac{3}{x + 6} + \frac{2 x + 2}{x^{2} + 2}$