Algebra Examples

$\frac{9 x^{2} + 36 x + 8}{(x + 4) (x^{2} + 5 x + 6)}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

Factor.

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Step 1.1.1

Factor $x^{2} + 5 x + 6$ using the AC method.

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Step 1.1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $6$ and whose sum is $5$ .

$2, 3$

Step 1.1.1.2

Write the factored form using these integers.

$\frac{9 x^{2} + 36 x + 8}{(x + 4) ((x + 2) (x + 3))}$

Step 1.1.2

Remove unnecessary parentheses.

$\frac{9 x^{2} + 36 x + 8}{(x + 4) (x + 2) (x + 3)}$

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 $A$ .

$\frac{A}{x + 4}$

Step 1.3

$\frac{A}{x + 4} + \frac{B}{x + 2}$

Step 1.4

$\frac{A}{x + 4} + \frac{B}{x + 2} + \frac{C}{x + 3}$

Step 1.5

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(x + 4) (x + 2) (x + 3)$ .

$\frac{(9 x^{2} + 36 x + 8) (x + 4) (x + 2) (x + 3)}{(x + 4) (x + 2) (x + 3)} = \frac{(A) (x + 4) (x + 2) (x + 3)}{x + 4} + \frac{(B) (x + 4) (x + 2) (x + 3)}{x + 2} + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.6

Cancel the common factor of $x + 4$ .

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Step 1.6.1

Cancel the common factor.

Step 1.6.2

Rewrite the expression.

$\frac{((9 x^{2} + 36 x + 8) (x + 2)) (x + 3)}{(x + 2) (x + 3)} = \frac{(A) (x + 4) (x + 2) (x + 3)}{x + 4} + \frac{(B) (x + 4) (x + 2) (x + 3)}{x + 2} + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.7

Cancel the common factor of $x + 2$ .

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Step 1.7.1

Cancel the common factor.

$\frac{(9 x^{2} + 36 x + 8) (x + 2) (x + 3)}{(x + 2) (x + 3)} = \frac{(A) (x + 4) (x + 2) (x + 3)}{x + 4} + \frac{(B) (x + 4) (x + 2) (x + 3)}{x + 2} + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.7.2

Rewrite the expression.

$\frac{(9 x^{2} + 36 x + 8) (x + 3)}{x + 3} = \frac{(A) (x + 4) (x + 2) (x + 3)}{x + 4} + \frac{(B) (x + 4) (x + 2) (x + 3)}{x + 2} + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.8

Cancel the common factor of $x + 3$ .

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Step 1.8.1

Cancel the common factor.

$\frac{(9 x^{2} + 36 x + 8) (x + 3)}{x + 3} = \frac{(A) (x + 4) (x + 2) (x + 3)}{x + 4} + \frac{(B) (x + 4) (x + 2) (x + 3)}{x + 2} + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.8.2

Divide $9 x^{2} + 36 x + 8$ by $1$ .

$9 x^{2} + 36 x + 8 = \frac{(A) (x + 4) (x + 2) (x + 3)}{x + 4} + \frac{(B) (x + 4) (x + 2) (x + 3)}{x + 2} + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9

Simplify each term.

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Step 1.9.1

Cancel the common factor of $x + 4$ .

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Step 1.9.1.1

Cancel the common factor.

$9 x^{2} + 36 x + 8 = \frac{A (x + 4) (x + 2) (x + 3)}{x + 4} + \frac{(B) (x + 4) (x + 2) (x + 3)}{x + 2} + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.1.2

Divide $((A) (x + 2)) (x + 3)$ by $1$ .

$9 x^{2} + 36 x + 8 = ((A) (x + 2)) (x + 3) + \frac{(B) (x + 4) (x + 2) (x + 3)}{x + 2} + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.2

Apply the distributive property.

$9 x^{2} + 36 x + 8 = (A x + A \cdot 2) (x + 3) + \frac{(B) (x + 4) (x + 2) (x + 3)}{x + 2} + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.3

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

$9 x^{2} + 36 x + 8 = (A x + 2 \cdot A) (x + 3) + \frac{(B) (x + 4) (x + 2) (x + 3)}{x + 2} + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.4

Expand $(A x + 2 A) (x + 3)$ using the FOIL Method.

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Step 1.9.4.1

Apply the distributive property.

$9 x^{2} + 36 x + 8 = A x (x + 3) + 2 A (x + 3) + \frac{(B) (x + 4) (x + 2) (x + 3)}{x + 2} + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.4.2

Apply the distributive property.

$9 x^{2} + 36 x + 8 = A x \cdot x + A x \cdot 3 + 2 A (x + 3) + \frac{(B) (x + 4) (x + 2) (x + 3)}{x + 2} + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.4.3

Apply the distributive property.

$9 x^{2} + 36 x + 8 = A x \cdot x + A x \cdot 3 + 2 A x + 2 A \cdot 3 + \frac{(B) (x + 4) (x + 2) (x + 3)}{x + 2} + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.5

Simplify and combine like terms.

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Step 1.9.5.1

Simplify each term.

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Step 1.9.5.1.1

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

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Step 1.9.5.1.1.1

Move $x$ .

$9 x^{2} + 36 x + 8 = A (x \cdot x) + A x \cdot 3 + 2 A x + 2 A \cdot 3 + \frac{(B) (x + 4) (x + 2) (x + 3)}{x + 2} + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.5.1.1.2

Multiply $x$ by $x$ .

$9 x^{2} + 36 x + 8 = A x^{2} + A x \cdot 3 + 2 A x + 2 A \cdot 3 + \frac{(B) (x + 4) (x + 2) (x + 3)}{x + 2} + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.5.1.2

Move $3$ to the left of $A x$ .

$9 x^{2} + 36 x + 8 = A x^{2} + 3 \cdot (A x) + 2 A x + 2 A \cdot 3 + \frac{(B) (x + 4) (x + 2) (x + 3)}{x + 2} + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.5.1.3

Multiply $3$ by $2$ .

$9 x^{2} + 36 x + 8 = A x^{2} + 3 A x + 2 A x + 6 A + \frac{(B) (x + 4) (x + 2) (x + 3)}{x + 2} + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.5.2

Add $3 A x$ and $2 A x$ .

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + \frac{(B) (x + 4) (x + 2) (x + 3)}{x + 2} + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.6

Cancel the common factor of $x + 2$ .

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Step 1.9.6.1

Cancel the common factor.

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + \frac{(B) (x + 4) (x + 2) (x + 3)}{x + 2} + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.6.2

Divide $(B) (x + 4) (x + 3)$ by $1$ .

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + (B) (x + 4) (x + 3) + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.7

Apply the distributive property.

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + (B x + B \cdot 4) (x + 3) + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.8

Move $4$ to the left of $B$ .

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + (B x + 4 \cdot B) (x + 3) + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.9

Expand $(B x + 4 B) (x + 3)$ using the FOIL Method.

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Step 1.9.9.1

Apply the distributive property.

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + B x (x + 3) + 4 B (x + 3) + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.9.2

Apply the distributive property.

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + B x \cdot x + B x \cdot 3 + 4 B (x + 3) + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.9.3

Apply the distributive property.

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + B x \cdot x + B x \cdot 3 + 4 B x + 4 B \cdot 3 + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.10

Simplify and combine like terms.

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Step 1.9.10.1

Simplify each term.

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Step 1.9.10.1.1

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

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Step 1.9.10.1.1.1

Move $x$ .

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + B (x \cdot x) + B x \cdot 3 + 4 B x + 4 B \cdot 3 + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.10.1.1.2

Multiply $x$ by $x$ .

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + B x^{2} + B x \cdot 3 + 4 B x + 4 B \cdot 3 + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.10.1.2

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

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + B x^{2} + 3 \cdot (B x) + 4 B x + 4 B \cdot 3 + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.10.1.3

Multiply $3$ by $4$ .

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + B x^{2} + 3 B x + 4 B x + 12 B + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.10.2

Add $3 B x$ and $4 B x$ .

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + B x^{2} + 7 B x + 12 B + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.11

Cancel the common factor of $x + 3$ .

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Step 1.9.11.1

Cancel the common factor.

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + B x^{2} + 7 B x + 12 B + \frac{(C) (x + 4) (x + 2) (x + 3)}{x + 3}$

Step 1.9.11.2

Divide $(C) (x + 4) (x + 2)$ by $1$ .

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + B x^{2} + 7 B x + 12 B + (C) (x + 4) (x + 2)$

Step 1.9.12

Apply the distributive property.

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + B x^{2} + 7 B x + 12 B + (C x + C \cdot 4) (x + 2)$

Step 1.9.13

Move $4$ to the left of $C$ .

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + B x^{2} + 7 B x + 12 B + (C x + 4 \cdot C) (x + 2)$

Step 1.9.14

Expand $(C x + 4 C) (x + 2)$ using the FOIL Method.

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Step 1.9.14.1

Apply the distributive property.

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + B x^{2} + 7 B x + 12 B + C x (x + 2) + 4 C (x + 2)$

Step 1.9.14.2

Apply the distributive property.

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + B x^{2} + 7 B x + 12 B + C x \cdot x + C x \cdot 2 + 4 C (x + 2)$

Step 1.9.14.3

Apply the distributive property.

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + B x^{2} + 7 B x + 12 B + C x \cdot x + C x \cdot 2 + 4 C x + 4 C \cdot 2$

Step 1.9.15

Simplify and combine like terms.

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Step 1.9.15.1

Simplify each term.

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Step 1.9.15.1.1

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

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Step 1.9.15.1.1.1

Move $x$ .

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + B x^{2} + 7 B x + 12 B + C (x \cdot x) + C x \cdot 2 + 4 C x + 4 C \cdot 2$

Step 1.9.15.1.1.2

Multiply $x$ by $x$ .

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + B x^{2} + 7 B x + 12 B + C x^{2} + C x \cdot 2 + 4 C x + 4 C \cdot 2$

Step 1.9.15.1.2

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

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + B x^{2} + 7 B x + 12 B + C x^{2} + 2 \cdot (C x) + 4 C x + 4 C \cdot 2$

Step 1.9.15.1.3

Multiply $2$ by $4$ .

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + B x^{2} + 7 B x + 12 B + C x^{2} + 2 C x + 4 C x + 8 C$

Step 1.9.15.2

Add $2 C x$ and $4 C x$ .

$9 x^{2} + 36 x + 8 = A x^{2} + 5 A x + 6 A + B x^{2} + 7 B x + 12 B + C x^{2} + 6 C x + 8 C$

Step 1.10

Simplify the expression.

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Step 1.10.1

Move $A$ .

$9 x^{2} + 36 x + 8 = A x^{2} + 5 x A + 6 A + B x^{2} + 7 B x + 12 B + C x^{2} + 6 C x + 8 C$

Step 1.10.2

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

$9 x^{2} + 36 x + 8 = A x^{2} + 5 x A + 6 A + B x^{2} + 7 B x + 12 B + C x^{2} + 6 C x + 8 C$

Step 1.10.3

Move $B$ .

$9 x^{2} + 36 x + 8 = A x^{2} + 5 x A + 6 A + B x^{2} + 7 x B + 12 B + C x^{2} + 6 C x + 8 C$

Step 1.10.4

Move $12 B$ .

$9 x^{2} + 36 x + 8 = A x^{2} + 5 x A + 6 A + B x^{2} + 7 x B + C x^{2} + 6 C x + 12 B + 8 C$

Step 1.10.5

Move $6 A$ .

$9 x^{2} + 36 x + 8 = A x^{2} + 5 x A + B x^{2} + 7 x B + C x^{2} + 6 C x + 6 A + 12 B + 8 C$

Step 1.10.6

Move $7 x B$ .

$9 x^{2} + 36 x + 8 = A x^{2} + 5 x A + B x^{2} + C x^{2} + 7 x B + 6 C x + 6 A + 12 B + 8 C$

Step 1.10.7

Move $5 x A$ .

$9 x^{2} + 36 x + 8 = A x^{2} + B x^{2} + C x^{2} + 5 x A + 7 x B + 6 C x + 6 A + 12 B + 8 C$

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.

$9 = A + B + C$

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.

$36 = 5 A + 7 B + 6 C$

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.

$8 = 6 A + 12 B + 8 C$

Step 2.4

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

$9 = A + B + C$

$36 = 5 A + 7 B + 6 C$

$8 = 6 A + 12 B + 8 C$

$9 = A + B + C$

$36 = 5 A + 7 B + 6 C$

$8 = 6 A + 12 B + 8 C$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $A$ in $9 = A + B + C$ .

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Step 3.1.1

Rewrite the equation as $A + B + C = 9$ .

$A + B + C = 9$

$36 = 5 A + 7 B + 6 C$

$8 = 6 A + 12 B + 8 C$

Step 3.1.2

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

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Step 3.1.2.1

Subtract $B$ from both sides of the equation.

$A + C = 9 - B$

$36 = 5 A + 7 B + 6 C$

$8 = 6 A + 12 B + 8 C$

Step 3.1.2.2

Subtract $C$ from both sides of the equation.

$A = 9 - B - C$

$36 = 5 A + 7 B + 6 C$

$8 = 6 A + 12 B + 8 C$

$A = 9 - B - C$

$36 = 5 A + 7 B + 6 C$

$8 = 6 A + 12 B + 8 C$

$A = 9 - B - C$

$36 = 5 A + 7 B + 6 C$

$8 = 6 A + 12 B + 8 C$

Step 3.2

Replace all occurrences of $A$ with $9 - B - C$ in each equation.

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Step 3.2.1

Replace all occurrences of $A$ in $36 = 5 A + 7 B + 6 C$ with $9 - B - C$ .

$36 = 5 (9 - B - C) + 7 B + 6 C$

$A = 9 - B - C$

$8 = 6 A + 12 B + 8 C$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $5 (9 - B - C) + 7 B + 6 C$ .

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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.

$36 = 5 \cdot 9 + 5 (- B) + 5 (- C) + 7 B + 6 C$

$A = 9 - B - C$

$8 = 6 A + 12 B + 8 C$

Step 3.2.2.1.1.2

Simplify.

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Step 3.2.2.1.1.2.1

Multiply $5$ by $9$ .

$36 = 45 + 5 (- B) + 5 (- C) + 7 B + 6 C$

$A = 9 - B - C$

$8 = 6 A + 12 B + 8 C$

Step 3.2.2.1.1.2.2

Multiply $- 1$ by $5$ .

$36 = 45 - 5 B + 5 (- C) + 7 B + 6 C$

$A = 9 - B - C$

$8 = 6 A + 12 B + 8 C$

Step 3.2.2.1.1.2.3

Multiply $- 1$ by $5$ .

$36 = 45 - 5 B - 5 C + 7 B + 6 C$

$A = 9 - B - C$

$8 = 6 A + 12 B + 8 C$

$36 = 45 - 5 B - 5 C + 7 B + 6 C$

$A = 9 - B - C$

$8 = 6 A + 12 B + 8 C$

$36 = 45 - 5 B - 5 C + 7 B + 6 C$

$A = 9 - B - C$

$8 = 6 A + 12 B + 8 C$

Step 3.2.2.1.2

Simplify by adding terms.

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Step 3.2.2.1.2.1

Add $- 5 B$ and $7 B$ .

$36 = 45 + 2 B - 5 C + 6 C$

$A = 9 - B - C$

$8 = 6 A + 12 B + 8 C$

Step 3.2.2.1.2.2

Add $- 5 C$ and $6 C$ .

$36 = 45 + 2 B + C$

$A = 9 - B - C$

$8 = 6 A + 12 B + 8 C$

$36 = 45 + 2 B + C$

$A = 9 - B - C$

$8 = 6 A + 12 B + 8 C$

$36 = 45 + 2 B + C$

$A = 9 - B - C$

$8 = 6 A + 12 B + 8 C$

$36 = 45 + 2 B + C$

$A = 9 - B - C$

$8 = 6 A + 12 B + 8 C$

Step 3.2.3

Replace all occurrences of $A$ in $8 = 6 A + 12 B + 8 C$ with $9 - B - C$ .

$8 = 6 (9 - B - C) + 12 B + 8 C$

$36 = 45 + 2 B + C$

$A = 9 - B - C$

Step 3.2.4

Simplify the right side.

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Step 3.2.4.1

Simplify $6 (9 - B - C) + 12 B + 8 C$ .

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Step 3.2.4.1.1

Simplify each term.

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Step 3.2.4.1.1.1

Apply the distributive property.

$8 = 6 \cdot 9 + 6 (- B) + 6 (- C) + 12 B + 8 C$

$36 = 45 + 2 B + C$

$A = 9 - B - C$

Step 3.2.4.1.1.2

Simplify.

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Step 3.2.4.1.1.2.1

Multiply $6$ by $9$ .

$8 = 54 + 6 (- B) + 6 (- C) + 12 B + 8 C$

$36 = 45 + 2 B + C$

$A = 9 - B - C$

Step 3.2.4.1.1.2.2

Multiply $- 1$ by $6$ .

$8 = 54 - 6 B + 6 (- C) + 12 B + 8 C$

$36 = 45 + 2 B + C$

$A = 9 - B - C$

Step 3.2.4.1.1.2.3

Multiply $- 1$ by $6$ .

$8 = 54 - 6 B - 6 C + 12 B + 8 C$

$36 = 45 + 2 B + C$

$A = 9 - B - C$

$8 = 54 - 6 B - 6 C + 12 B + 8 C$

$36 = 45 + 2 B + C$

$A = 9 - B - C$

$8 = 54 - 6 B - 6 C + 12 B + 8 C$

$36 = 45 + 2 B + C$

$A = 9 - B - C$

Step 3.2.4.1.2

Simplify by adding terms.

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Step 3.2.4.1.2.1

Add $- 6 B$ and $12 B$ .

$8 = 54 + 6 B - 6 C + 8 C$

$36 = 45 + 2 B + C$

$A = 9 - B - C$

Step 3.2.4.1.2.2

Add $- 6 C$ and $8 C$ .

$8 = 54 + 6 B + 2 C$

$36 = 45 + 2 B + C$

$A = 9 - B - C$

$8 = 54 + 6 B + 2 C$

$36 = 45 + 2 B + C$

$A = 9 - B - C$

$8 = 54 + 6 B + 2 C$

$36 = 45 + 2 B + C$

$A = 9 - B - C$

$8 = 54 + 6 B + 2 C$

$36 = 45 + 2 B + C$

$A = 9 - B - C$

$8 = 54 + 6 B + 2 C$

$36 = 45 + 2 B + C$

$A = 9 - B - C$

Step 3.3

Solve for $C$ in $36 = 45 + 2 B + C$ .

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Step 3.3.1

Rewrite the equation as $45 + 2 B + C = 36$ .

$45 + 2 B + C = 36$

$8 = 54 + 6 B + 2 C$

$A = 9 - B - C$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $45$ from both sides of the equation.

$2 B + C = 36 - 45$

$8 = 54 + 6 B + 2 C$

$A = 9 - B - C$

Step 3.3.2.2

Subtract $2 B$ from both sides of the equation.

$C = 36 - 45 - 2 B$

$8 = 54 + 6 B + 2 C$

$A = 9 - B - C$

Step 3.3.2.3

Subtract $45$ from $36$ .

$C = - 9 - 2 B$

$8 = 54 + 6 B + 2 C$

$A = 9 - B - C$

$C = - 9 - 2 B$

$8 = 54 + 6 B + 2 C$

$A = 9 - B - C$

$C = - 9 - 2 B$

$8 = 54 + 6 B + 2 C$

$A = 9 - B - C$

Step 3.4

Replace all occurrences of $C$ with $- 9 - 2 B$ in each equation.

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Step 3.4.1

Replace all occurrences of $C$ in $8 = 54 + 6 B + 2 C$ with $- 9 - 2 B$ .

$8 = 54 + 6 B + 2 (- 9 - 2 B)$

$C = - 9 - 2 B$

$A = 9 - B - C$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $54 + 6 B + 2 (- 9 - 2 B)$ .

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Step 3.4.2.1.1

Simplify each term.

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Step 3.4.2.1.1.1

Apply the distributive property.

$8 = 54 + 6 B + 2 \cdot - 9 + 2 (- 2 B)$

$C = - 9 - 2 B$

$A = 9 - B - C$

Step 3.4.2.1.1.2

Multiply $2$ by $- 9$ .

$8 = 54 + 6 B - 18 + 2 (- 2 B)$

$C = - 9 - 2 B$

$A = 9 - B - C$

Step 3.4.2.1.1.3

Multiply $- 2$ by $2$ .

$8 = 54 + 6 B - 18 - 4 B$

$C = - 9 - 2 B$

$A = 9 - B - C$

$8 = 54 + 6 B - 18 - 4 B$

$C = - 9 - 2 B$

$A = 9 - B - C$

Step 3.4.2.1.2

Simplify by adding terms.

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Step 3.4.2.1.2.1

Subtract $18$ from $54$ .

$8 = 6 B + 36 - 4 B$

$C = - 9 - 2 B$

$A = 9 - B - C$

Step 3.4.2.1.2.2

Subtract $4 B$ from $6 B$ .

$8 = 2 B + 36$

$C = - 9 - 2 B$

$A = 9 - B - C$

$8 = 2 B + 36$

$C = - 9 - 2 B$

$A = 9 - B - C$

$8 = 2 B + 36$

$C = - 9 - 2 B$

$A = 9 - B - C$

$8 = 2 B + 36$

$C = - 9 - 2 B$

$A = 9 - B - C$

Step 3.4.3

Replace all occurrences of $C$ in $A = 9 - B - C$ with $- 9 - 2 B$ .

$A = 9 - B - (- 9 - 2 B)$

$8 = 2 B + 36$

$C = - 9 - 2 B$

Step 3.4.4

Simplify the right side.

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Step 3.4.4.1

Simplify $9 - B - (- 9 - 2 B)$ .

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Step 3.4.4.1.1

Simplify each term.

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Step 3.4.4.1.1.1

Apply the distributive property.

$A = 9 - B + 9 - (- 2 B)$

$8 = 2 B + 36$

$C = - 9 - 2 B$

Step 3.4.4.1.1.2

Multiply $- 1$ by $- 9$ .

$A = 9 - B + 9 - (- 2 B)$

$8 = 2 B + 36$

$C = - 9 - 2 B$

Step 3.4.4.1.1.3

Multiply $- 2$ by $- 1$ .

$A = 9 - B + 9 + 2 B$

$8 = 2 B + 36$

$C = - 9 - 2 B$

$A = 9 - B + 9 + 2 B$

$8 = 2 B + 36$

$C = - 9 - 2 B$

Step 3.4.4.1.2

Simplify by adding terms.

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Step 3.4.4.1.2.1

Add $9$ and $9$ .

$A = - B + 18 + 2 B$

$8 = 2 B + 36$

$C = - 9 - 2 B$

Step 3.4.4.1.2.2

Add $- B$ and $2 B$ .

$A = B + 18$

$8 = 2 B + 36$

$C = - 9 - 2 B$

$A = B + 18$

$8 = 2 B + 36$

$C = - 9 - 2 B$

$A = B + 18$

$8 = 2 B + 36$

$C = - 9 - 2 B$

$A = B + 18$

$8 = 2 B + 36$

$C = - 9 - 2 B$

$A = B + 18$

$8 = 2 B + 36$

$C = - 9 - 2 B$

Step 3.5

Solve for $B$ in $8 = 2 B + 36$ .

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Step 3.5.1

Rewrite the equation as $2 B + 36 = 8$ .

$2 B + 36 = 8$

$A = B + 18$

$C = - 9 - 2 B$

Step 3.5.2

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

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Step 3.5.2.1

Subtract $36$ from both sides of the equation.

$2 B = 8 - 36$

$A = B + 18$

$C = - 9 - 2 B$

Step 3.5.2.2

Subtract $36$ from $8$ .

$2 B = - 28$

$A = B + 18$

$C = - 9 - 2 B$

$2 B = - 28$

$A = B + 18$

$C = - 9 - 2 B$

Step 3.5.3

Divide each term in $2 B = - 28$ by $2$ and simplify.

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Step 3.5.3.1

Divide each term in $2 B = - 28$ by $2$ .

$\frac{2 B}{2} = \frac{- 28}{2}$

$A = B + 18$

$C = - 9 - 2 B$

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $2$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

$\frac{2 B}{2} = \frac{- 28}{2}$

$A = B + 18$

$C = - 9 - 2 B$

Step 3.5.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{- 28}{2}$

$A = B + 18$

$C = - 9 - 2 B$

$B = \frac{- 28}{2}$

$A = B + 18$

$C = - 9 - 2 B$

$B = \frac{- 28}{2}$

$A = B + 18$

$C = - 9 - 2 B$

Step 3.5.3.3

Simplify the right side.

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Step 3.5.3.3.1

Divide $- 28$ by $2$ .

$B = - 14$

$A = B + 18$

$C = - 9 - 2 B$

$B = - 14$

$A = B + 18$

$C = - 9 - 2 B$

$B = - 14$

$A = B + 18$

$C = - 9 - 2 B$

$B = - 14$

$A = B + 18$

$C = - 9 - 2 B$

Step 3.6

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

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Step 3.6.1

Replace all occurrences of $B$ in $A = B + 18$ with $- 14$ .

$A = (- 14) + 18$

$B = - 14$

$C = - 9 - 2 B$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Add $- 14$ and $18$ .

$A = 4$

$B = - 14$

$C = - 9 - 2 B$

$A = 4$

$B = - 14$

$C = - 9 - 2 B$

Step 3.6.3

Replace all occurrences of $B$ in $C = - 9 - 2 B$ with $- 14$ .

$C = - 9 - 2 \cdot - 14$

$A = 4$

$B = - 14$

Step 3.6.4

Simplify the right side.

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Step 3.6.4.1

Simplify $- 9 - 2 \cdot - 14$ .

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Step 3.6.4.1.1

Multiply $- 2$ by $- 14$ .

$C = - 9 + 28$

$A = 4$

$B = - 14$

Step 3.6.4.1.2

Add $- 9$ and $28$ .

$C = 19$

$A = 4$

$B = - 14$

$C = 19$

$A = 4$

$B = - 14$

$C = 19$

$A = 4$

$B = - 14$

$C = 19$

$A = 4$

$B = - 14$

Step 3.7

List all of the solutions.

$C = 19, A = 4, B = - 14$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x + 4} + \frac{B}{x + 2} + \frac{C}{x + 3}$ with the values found for $A$ , $B$ , and $C$ .

$\frac{4}{x + 4} + \frac{- 14}{x + 2} + \frac{19}{x + 3}$