Precalculus Examples

$\frac{x^{2} + 4 x + 2}{(x + 2) (x^{2} + 22 x + 6)}$

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 $A$ .

$\frac{A}{x + 2}$

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.

$\frac{A}{x + 2} + \frac{B x + C}{x^{2} + 22 x + 6}$

Step 1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(x + 2) (x^{2} + 22 x + 6)$ .

$\frac{(x^{2} + 4 x + 2) (x + 2) (x^{2} + 22 x + 6)}{(x + 2) (x^{2} + 22 x + 6)} = \frac{(A) (x + 2) (x^{2} + 22 x + 6)}{x + 2} + \frac{(B x + C) (x + 2) (x^{2} + 22 x + 6)}{x^{2} + 22 x + 6}$

Step 1.4

Cancel the common factor of $x + 2$ .

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Step 1.4.1

Cancel the common factor.

$\frac{(x^{2} + 4 x + 2) (x + 2) (x^{2} + 22 x + 6)}{(x + 2) (x^{2} + 22 x + 6)} = \frac{(A) (x + 2) (x^{2} + 22 x + 6)}{x + 2} + \frac{(B x + C) (x + 2) (x^{2} + 22 x + 6)}{x^{2} + 22 x + 6}$

Step 1.4.2

Rewrite the expression.

$\frac{(x^{2} + 4 x + 2) (x^{2} + 22 x + 6)}{x^{2} + 22 x + 6} = \frac{(A) (x + 2) (x^{2} + 22 x + 6)}{x + 2} + \frac{(B x + C) (x + 2) (x^{2} + 22 x + 6)}{x^{2} + 22 x + 6}$

Step 1.5

Cancel the common factor of $x^{2} + 22 x + 6$ .

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Step 1.5.1

Cancel the common factor.

$\frac{(x^{2} + 4 x + 2) (x^{2} + 22 x + 6)}{x^{2} + 22 x + 6} = \frac{(A) (x + 2) (x^{2} + 22 x + 6)}{x + 2} + \frac{(B x + C) (x + 2) (x^{2} + 22 x + 6)}{x^{2} + 22 x + 6}$

Step 1.5.2

Divide $x^{2} + 4 x + 2$ by $1$ .

$x^{2} + 4 x + 2 = \frac{(A) (x + 2) (x^{2} + 22 x + 6)}{x + 2} + \frac{(B x + C) (x + 2) (x^{2} + 22 x + 6)}{x^{2} + 22 x + 6}$

Step 1.6

Simplify each term.

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Step 1.6.1

Cancel the common factor of $x + 2$ .

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Step 1.6.1.1

Cancel the common factor.

$x^{2} + 4 x + 2 = \frac{A (x + 2) (x^{2} + 22 x + 6)}{x + 2} + \frac{(B x + C) (x + 2) (x^{2} + 22 x + 6)}{x^{2} + 22 x + 6}$

Step 1.6.1.2

Divide $(A) (x^{2} + 22 x + 6)$ by $1$ .

$x^{2} + 4 x + 2 = (A) (x^{2} + 22 x + 6) + \frac{(B x + C) (x + 2) (x^{2} + 22 x + 6)}{x^{2} + 22 x + 6}$

Step 1.6.2

Apply the distributive property.

$x^{2} + 4 x + 2 = A x^{2} + A (22 x) + A \cdot 6 + \frac{(B x + C) (x + 2) (x^{2} + 22 x + 6)}{x^{2} + 22 x + 6}$

Step 1.6.3

Simplify.

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Step 1.6.3.1

Rewrite using the commutative property of multiplication.

$x^{2} + 4 x + 2 = A x^{2} + 22 A x + A \cdot 6 + \frac{(B x + C) (x + 2) (x^{2} + 22 x + 6)}{x^{2} + 22 x + 6}$

Step 1.6.3.2

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

$x^{2} + 4 x + 2 = A x^{2} + 22 A x + 6 \cdot A + \frac{(B x + C) (x + 2) (x^{2} + 22 x + 6)}{x^{2} + 22 x + 6}$

Step 1.6.4

Cancel the common factor of $x^{2} + 22 x + 6$ .

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Step 1.6.4.1

Cancel the common factor.

$x^{2} + 4 x + 2 = A x^{2} + 22 A x + 6 A + \frac{(B x + C) (x + 2) (x^{2} + 22 x + 6)}{x^{2} + 22 x + 6}$

Step 1.6.4.2

Divide $(B x + C) (x + 2)$ by $1$ .

$x^{2} + 4 x + 2 = A x^{2} + 22 A x + 6 A + (B x + C) (x + 2)$

Step 1.6.5

Expand $(B x + C) (x + 2)$ using the FOIL Method.

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Step 1.6.5.1

Apply the distributive property.

$x^{2} + 4 x + 2 = A x^{2} + 22 A x + 6 A + B x (x + 2) + C (x + 2)$

Step 1.6.5.2

Apply the distributive property.

$x^{2} + 4 x + 2 = A x^{2} + 22 A x + 6 A + B x \cdot x + B x \cdot 2 + C (x + 2)$

Step 1.6.5.3

Apply the distributive property.

$x^{2} + 4 x + 2 = A x^{2} + 22 A x + 6 A + B x \cdot x + B x \cdot 2 + C x + C \cdot 2$

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 $x$ .

$x^{2} + 4 x + 2 = A x^{2} + 22 A x + 6 A + B (x \cdot x) + B x \cdot 2 + C x + C \cdot 2$

Step 1.6.6.1.2

Multiply $x$ by $x$ .

$x^{2} + 4 x + 2 = A x^{2} + 22 A x + 6 A + B x^{2} + B x \cdot 2 + C x + C \cdot 2$

Step 1.6.6.2

Move $2$ to the left of $B x$ .

$x^{2} + 4 x + 2 = A x^{2} + 22 A x + 6 A + B x^{2} + 2 \cdot (B x) + C x + C \cdot 2$

Step 1.6.6.3

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

$x^{2} + 4 x + 2 = A x^{2} + 22 A x + 6 A + B x^{2} + 2 B x + C x + 2 C$

Step 1.7

Simplify the expression.

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Step 1.7.1

Move $A$ .

$x^{2} + 4 x + 2 = A x^{2} + 22 x A + 6 A + B x^{2} + 2 B x + C x + 2 C$

Step 1.7.2

Reorder $B$ and $x^{2}$ .

$x^{2} + 4 x + 2 = A x^{2} + 22 x A + 6 A + B x^{2} + 2 B x + C x + 2 C$

Step 1.7.3

Move $B$ .

$x^{2} + 4 x + 2 = A x^{2} + 22 x A + 6 A + B x^{2} + 2 x B + C x + 2 C$

Step 1.7.4

Move $6 A$ .

$x^{2} + 4 x + 2 = A x^{2} + 22 x A + B x^{2} + 2 x B + C x + 6 A + 2 C$

Step 1.7.5

Move $22 x A$ .

$x^{2} + 4 x + 2 = A x^{2} + B x^{2} + 22 x A + 2 x B + C x + 6 A + 2 C$

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.

$1 = A + B$

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.

$4 = 22 A + 2 B + C$

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.

$2 = 6 A + 2 C$

Step 2.4

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

$1 = A + B$

$4 = 22 A + 2 B + C$

$2 = 6 A + 2 C$

$1 = A + B$

$4 = 22 A + 2 B + C$

$2 = 6 A + 2 C$

Step 3

Solve the system of equations.

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Step 3.1

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

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Step 3.1.1

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

$A + B = 1$

$4 = 22 A + 2 B + C$

$2 = 6 A + 2 C$

Step 3.1.2

Subtract $B$ from both sides of the equation.

$A = 1 - B$

$4 = 22 A + 2 B + C$

$2 = 6 A + 2 C$

$A = 1 - B$

$4 = 22 A + 2 B + C$

$2 = 6 A + 2 C$

Step 3.2

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

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Step 3.2.1

Replace all occurrences of $A$ in $4 = 22 A + 2 B + C$ with $1 - B$ .

$4 = 22 (1 - B) + 2 B + C$

$A = 1 - B$

$2 = 6 A + 2 C$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $22 (1 - B) + 2 B + C$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Apply the distributive property.

$4 = 22 \cdot 1 + 22 (- B) + 2 B + C$

$A = 1 - B$

$2 = 6 A + 2 C$

Step 3.2.2.1.1.2

Multiply $22$ by $1$ .

$4 = 22 + 22 (- B) + 2 B + C$

$A = 1 - B$

$2 = 6 A + 2 C$

Step 3.2.2.1.1.3

Multiply $- 1$ by $22$ .

$4 = 22 - 22 B + 2 B + C$

$A = 1 - B$

$2 = 6 A + 2 C$

$4 = 22 - 22 B + 2 B + C$

$A = 1 - B$

$2 = 6 A + 2 C$

Step 3.2.2.1.2

Add $- 22 B$ and $2 B$ .

$4 = 22 - 20 B + C$

$A = 1 - B$

$2 = 6 A + 2 C$

$4 = 22 - 20 B + C$

$A = 1 - B$

$2 = 6 A + 2 C$

$4 = 22 - 20 B + C$

$A = 1 - B$

$2 = 6 A + 2 C$

Step 3.2.3

Replace all occurrences of $A$ in $2 = 6 A + 2 C$ with $1 - B$ .

$2 = 6 (1 - B) + 2 C$

$4 = 22 - 20 B + C$

$A = 1 - B$

Step 3.2.4

Simplify the right side.

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Step 3.2.4.1

Simplify each term.

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Step 3.2.4.1.1

Apply the distributive property.

$2 = 6 \cdot 1 + 6 (- B) + 2 C$

$4 = 22 - 20 B + C$

$A = 1 - B$

Step 3.2.4.1.2

Multiply $6$ by $1$ .

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

$4 = 22 - 20 B + C$

$A = 1 - B$

Step 3.2.4.1.3

Multiply $- 1$ by $6$ .

$2 = 6 - 6 B + 2 C$

$4 = 22 - 20 B + C$

$A = 1 - B$

$2 = 6 - 6 B + 2 C$

$4 = 22 - 20 B + C$

$A = 1 - B$

$2 = 6 - 6 B + 2 C$

$4 = 22 - 20 B + C$

$A = 1 - B$

$2 = 6 - 6 B + 2 C$

$4 = 22 - 20 B + C$

$A = 1 - B$

Step 3.3

Reorder $1$ and $- B$ .

$A = - B + 1$

$2 = 6 - 6 B + 2 C$

$4 = 22 - 20 B + C$

Step 3.4

Solve for $C$ in $4 = 22 - 20 B + C$ .

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Step 3.4.1

Rewrite the equation as $22 - 20 B + C = 4$ .

$22 - 20 B + C = 4$

$A = - B + 1$

$2 = 6 - 6 B + 2 C$

Step 3.4.2

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

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Step 3.4.2.1

Subtract $22$ from both sides of the equation.

$- 20 B + C = 4 - 22$

$A = - B + 1$

$2 = 6 - 6 B + 2 C$

Step 3.4.2.2

Add $20 B$ to both sides of the equation.

$C = 4 - 22 + 20 B$

$A = - B + 1$

$2 = 6 - 6 B + 2 C$

Step 3.4.2.3

Subtract $22$ from $4$ .

$C = - 18 + 20 B$

$A = - B + 1$

$2 = 6 - 6 B + 2 C$

$C = - 18 + 20 B$

$A = - B + 1$

$2 = 6 - 6 B + 2 C$

$C = - 18 + 20 B$

$A = - B + 1$

$2 = 6 - 6 B + 2 C$

Step 3.5

Replace all occurrences of $C$ with $- 18 + 20 B$ in each equation.

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Step 3.5.1

Replace all occurrences of $C$ in $2 = 6 - 6 B + 2 C$ with $- 18 + 20 B$ .

$2 = 6 - 6 B + 2 (- 18 + 20 B)$

$C = - 18 + 20 B$

$A = - B + 1$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify $6 - 6 B + 2 (- 18 + 20 B)$ .

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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.

$2 = 6 - 6 B + 2 \cdot - 18 + 2 (20 B)$

$C = - 18 + 20 B$

$A = - B + 1$

Step 3.5.2.1.1.2

Multiply $2$ by $- 18$ .

$2 = 6 - 6 B - 36 + 2 (20 B)$

$C = - 18 + 20 B$

$A = - B + 1$

Step 3.5.2.1.1.3

Multiply $20$ by $2$ .

$2 = 6 - 6 B - 36 + 40 B$

$C = - 18 + 20 B$

$A = - B + 1$

$2 = 6 - 6 B - 36 + 40 B$

$C = - 18 + 20 B$

$A = - B + 1$

Step 3.5.2.1.2

Simplify by adding terms.

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Step 3.5.2.1.2.1

Subtract $36$ from $6$ .

$2 = - 6 B - 30 + 40 B$

$C = - 18 + 20 B$

$A = - B + 1$

Step 3.5.2.1.2.2

Add $- 6 B$ and $40 B$ .

$2 = 34 B - 30$

$C = - 18 + 20 B$

$A = - B + 1$

$2 = 34 B - 30$

$C = - 18 + 20 B$

$A = - B + 1$

$2 = 34 B - 30$

$C = - 18 + 20 B$

$A = - B + 1$

$2 = 34 B - 30$

$C = - 18 + 20 B$

$A = - B + 1$

$2 = 34 B - 30$

$C = - 18 + 20 B$

$A = - B + 1$

Step 3.6

Solve for $B$ in $2 = 34 B - 30$ .

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Step 3.6.1

Rewrite the equation as $34 B - 30 = 2$ .

$34 B - 30 = 2$

$C = - 18 + 20 B$

$A = - B + 1$

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

Add $30$ to both sides of the equation.

$34 B = 2 + 30$

$C = - 18 + 20 B$

$A = - B + 1$

Step 3.6.2.2

Add $2$ and $30$ .

$34 B = 32$

$C = - 18 + 20 B$

$A = - B + 1$

$34 B = 32$

$C = - 18 + 20 B$

$A = - B + 1$

Step 3.6.3

Divide each term in $34 B = 32$ by $34$ and simplify.

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Step 3.6.3.1

Divide each term in $34 B = 32$ by $34$ .

$\frac{34 B}{34} = \frac{32}{34}$

$C = - 18 + 20 B$

$A = - B + 1$

Step 3.6.3.2

Simplify the left side.

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Step 3.6.3.2.1

Cancel the common factor of $34$ .

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Step 3.6.3.2.1.1

Cancel the common factor.

$\frac{34 B}{34} = \frac{32}{34}$

$C = - 18 + 20 B$

$A = - B + 1$

Step 3.6.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{32}{34}$

$C = - 18 + 20 B$

$A = - B + 1$

$B = \frac{32}{34}$

$C = - 18 + 20 B$

$A = - B + 1$

$B = \frac{32}{34}$

$C = - 18 + 20 B$

$A = - B + 1$

Step 3.6.3.3

Simplify the right side.

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Step 3.6.3.3.1

Cancel the common factor of $32$ and $34$ .

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Step 3.6.3.3.1.1

Factor $2$ out of $32$ .

$B = \frac{2 (16)}{34}$

$C = - 18 + 20 B$

$A = - B + 1$

Step 3.6.3.3.1.2

Cancel the common factors.

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Step 3.6.3.3.1.2.1

Factor $2$ out of $34$ .

$B = \frac{2 \cdot 16}{2 \cdot 17}$

$C = - 18 + 20 B$

$A = - B + 1$

Step 3.6.3.3.1.2.2

Cancel the common factor.

$B = \frac{2 \cdot 16}{2 \cdot 17}$

$C = - 18 + 20 B$

$A = - B + 1$

Step 3.6.3.3.1.2.3

Rewrite the expression.

$B = \frac{16}{17}$

$C = - 18 + 20 B$

$A = - B + 1$

$B = \frac{16}{17}$

$C = - 18 + 20 B$

$A = - B + 1$

$B = \frac{16}{17}$

$C = - 18 + 20 B$

$A = - B + 1$

$B = \frac{16}{17}$

$C = - 18 + 20 B$

$A = - B + 1$

$B = \frac{16}{17}$

$C = - 18 + 20 B$

$A = - B + 1$

$B = \frac{16}{17}$

$C = - 18 + 20 B$

$A = - B + 1$

Step 3.7

Replace all occurrences of $B$ with $\frac{16}{17}$ in each equation.

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Step 3.7.1

Replace all occurrences of $B$ in $C = - 18 + 20 B$ with $\frac{16}{17}$ .

$C = - 18 + 20 (\frac{16}{17})$

$B = \frac{16}{17}$

$A = - B + 1$

Step 3.7.2

Simplify the right side.

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Step 3.7.2.1

Simplify $- 18 + 20 (\frac{16}{17})$ .

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Step 3.7.2.1.1

Multiply $20 (\frac{16}{17})$ .

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Step 3.7.2.1.1.1

Combine $20$ and $\frac{16}{17}$ .

$C = - 18 + \frac{20 \cdot 16}{17}$

$B = \frac{16}{17}$

$A = - B + 1$

Step 3.7.2.1.1.2

Multiply $20$ by $16$ .

$C = - 18 + \frac{320}{17}$

$B = \frac{16}{17}$

$A = - B + 1$

$C = - 18 + \frac{320}{17}$

$B = \frac{16}{17}$

$A = - B + 1$

Step 3.7.2.1.2

To write $- 18$ as a fraction with a common denominator, multiply by $\frac{17}{17}$ .

$C = - 18 \cdot \frac{17}{17} + \frac{320}{17}$

$B = \frac{16}{17}$

$A = - B + 1$

Step 3.7.2.1.3

Combine $- 18$ and $\frac{17}{17}$ .

$C = \frac{- 18 \cdot 17}{17} + \frac{320}{17}$

$B = \frac{16}{17}$

$A = - B + 1$

Step 3.7.2.1.4

Combine the numerators over the common denominator.

$C = \frac{- 18 \cdot 17 + 320}{17}$

$B = \frac{16}{17}$

$A = - B + 1$

Step 3.7.2.1.5

Simplify the numerator.

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Step 3.7.2.1.5.1

Multiply $- 18$ by $17$ .

$C = \frac{- 306 + 320}{17}$

$B = \frac{16}{17}$

$A = - B + 1$

Step 3.7.2.1.5.2

Add $- 306$ and $320$ .

$C = \frac{14}{17}$

$B = \frac{16}{17}$

$A = - B + 1$

$C = \frac{14}{17}$

$B = \frac{16}{17}$

$A = - B + 1$

$C = \frac{14}{17}$

$B = \frac{16}{17}$

$A = - B + 1$

$C = \frac{14}{17}$

$B = \frac{16}{17}$

$A = - B + 1$

Step 3.7.3

Replace all occurrences of $B$ in $A = - B + 1$ with $\frac{16}{17}$ .

$A = - (\frac{16}{17}) + 1$

$C = \frac{14}{17}$

$B = \frac{16}{17}$

Step 3.7.4

Simplify the right side.

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Step 3.7.4.1

Simplify $- (\frac{16}{17}) + 1$ .

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Step 3.7.4.1.1

Write $1$ as a fraction with a common denominator.

$A = - \frac{16}{17} + \frac{17}{17}$

$C = \frac{14}{17}$

$B = \frac{16}{17}$

Step 3.7.4.1.2

Combine the numerators over the common denominator.

$A = \frac{- 16 + 17}{17}$

$C = \frac{14}{17}$

$B = \frac{16}{17}$

Step 3.7.4.1.3

Add $- 16$ and $17$ .

$A = \frac{1}{17}$

$C = \frac{14}{17}$

$B = \frac{16}{17}$

$A = \frac{1}{17}$

$C = \frac{14}{17}$

$B = \frac{16}{17}$

$A = \frac{1}{17}$

$C = \frac{14}{17}$

$B = \frac{16}{17}$

$A = \frac{1}{17}$

$C = \frac{14}{17}$

$B = \frac{16}{17}$

Step 3.8

List all of the solutions.

$A = \frac{1}{17}, C = \frac{14}{17}, B = \frac{16}{17}$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x + 2} + \frac{B x + C}{x^{2} + 22 x + 6}$ with the values found for $A$ , $B$ , and $C$ .

$\frac{\frac{1}{17}}{x + 2} + \frac{\frac{16}{17 x} + \frac{14}{17}}{x^{2} + 22 x + 6}$

Step 5

Combine $\frac{16}{17}$ and $x$ .

$\frac{\frac{1}{17}}{x + 2} + \frac{\frac{16 x}{17} + \frac{14}{17}}{x^{2} + 22 x + 6}$