Precalculus Examples

$\frac{5 x + 69}{{(x - 3)}^{2} (x + 4)}$

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 $A$ .

$\frac{A}{{(x - 3)}^{2}}$

Step 1.2

$\frac{A}{{(x - 3)}^{2}} + \frac{B}{x - 3}$

Step 1.3

$\frac{A}{{(x - 3)}^{2}} + \frac{B}{x - 3} + \frac{C}{x + 4}$

Step 1.4

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

$\frac{(5 x + 69) {(x - 3)}^{2} (x + 4)}{{(x - 3)}^{2} (x + 4)} = \frac{(A) {(x - 3)}^{2} (x + 4)}{{(x - 3)}^{2}} + \frac{(B) {(x - 3)}^{2} (x + 4)}{x - 3} + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.5

Cancel the common factor of ${(x - 3)}^{2}$ .

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Step 1.5.1

Cancel the common factor.

$\frac{(5 x + 69) {(x - 3)}^{2} (x + 4)}{{(x - 3)}^{2} (x + 4)} = \frac{(A) {(x - 3)}^{2} (x + 4)}{{(x - 3)}^{2}} + \frac{(B) {(x - 3)}^{2} (x + 4)}{x - 3} + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.5.2

Rewrite the expression.

$\frac{(5 x + 69) (x + 4)}{x + 4} = \frac{(A) {(x - 3)}^{2} (x + 4)}{{(x - 3)}^{2}} + \frac{(B) {(x - 3)}^{2} (x + 4)}{x - 3} + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.6

Cancel the common factor of $x + 4$ .

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Step 1.6.1

Cancel the common factor.

$\frac{(5 x + 69) (x + 4)}{x + 4} = \frac{(A) {(x - 3)}^{2} (x + 4)}{{(x - 3)}^{2}} + \frac{(B) {(x - 3)}^{2} (x + 4)}{x - 3} + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.6.2

Divide $5 x + 69$ by $1$ .

$5 x + 69 = \frac{(A) {(x - 3)}^{2} (x + 4)}{{(x - 3)}^{2}} + \frac{(B) {(x - 3)}^{2} (x + 4)}{x - 3} + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7

Simplify each term.

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Step 1.7.1

Cancel the common factor of ${(x - 3)}^{2}$ .

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Step 1.7.1.1

Cancel the common factor.

$5 x + 69 = \frac{A {(x - 3)}^{2} (x + 4)}{{(x - 3)}^{2}} + \frac{(B) {(x - 3)}^{2} (x + 4)}{x - 3} + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7.1.2

Divide $(A) (x + 4)$ by $1$ .

$5 x + 69 = (A) (x + 4) + \frac{(B) {(x - 3)}^{2} (x + 4)}{x - 3} + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7.2

Apply the distributive property.

$5 x + 69 = A x + A \cdot 4 + \frac{(B) {(x - 3)}^{2} (x + 4)}{x - 3} + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7.3

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

$5 x + 69 = A x + 4 \cdot A + \frac{(B) {(x - 3)}^{2} (x + 4)}{x - 3} + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7.4

Cancel the common factor of ${(x - 3)}^{2}$ and $x - 3$ .

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Step 1.7.4.1

Factor $x - 3$ out of $(B) {(x - 3)}^{2} (x + 4)$ .

$5 x + 69 = A x + 4 A + \frac{(x - 3) (B (x - 3) (x + 4))}{x - 3} + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7.4.2

Cancel the common factors.

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Step 1.7.4.2.1

Multiply by $1$ .

$5 x + 69 = A x + 4 A + \frac{(x - 3) (B (x - 3) (x + 4))}{(x - 3) \cdot 1} + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7.4.2.2

Cancel the common factor.

$5 x + 69 = A x + 4 A + \frac{(x - 3) (B (x - 3) (x + 4))}{(x - 3) \cdot 1} + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7.4.2.3

Rewrite the expression.

$5 x + 69 = A x + 4 A + \frac{B (x - 3) (x + 4)}{1} + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7.4.2.4

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

$5 x + 69 = A x + 4 A + B (x - 3) (x + 4) + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7.5

Apply the distributive property.

$5 x + 69 = A x + 4 A + (B x + B \cdot - 3) (x + 4) + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7.6

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

$5 x + 69 = A x + 4 A + (B x - 3 \cdot B) (x + 4) + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7.7

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

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Step 1.7.7.1

Apply the distributive property.

$5 x + 69 = A x + 4 A + B x (x + 4) - 3 B (x + 4) + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7.7.2

Apply the distributive property.

$5 x + 69 = A x + 4 A + B x \cdot x + B x \cdot 4 - 3 B (x + 4) + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7.7.3

Apply the distributive property.

$5 x + 69 = A x + 4 A + B x \cdot x + B x \cdot 4 - 3 B x - 3 B \cdot 4 + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7.8

Simplify and combine like terms.

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Step 1.7.8.1

Simplify each term.

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Step 1.7.8.1.1

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

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Step 1.7.8.1.1.1

Move $x$ .

$5 x + 69 = A x + 4 A + B (x \cdot x) + B x \cdot 4 - 3 B x - 3 B \cdot 4 + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7.8.1.1.2

Multiply $x$ by $x$ .

$5 x + 69 = A x + 4 A + B x^{2} + B x \cdot 4 - 3 B x - 3 B \cdot 4 + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7.8.1.2

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

$5 x + 69 = A x + 4 A + B x^{2} + 4 \cdot (B x) - 3 B x - 3 B \cdot 4 + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7.8.1.3

Multiply $4$ by $- 3$ .

$5 x + 69 = A x + 4 A + B x^{2} + 4 B x - 3 B x - 12 B + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7.8.2

Subtract $3 B x$ from $4 B x$ .

$5 x + 69 = A x + 4 A + B x^{2} + 1 B x - 12 B + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7.9

Multiply $B$ by $1$ .

$5 x + 69 = A x + 4 A + B x^{2} + B x - 12 B + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7.10

Cancel the common factor of $x + 4$ .

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Step 1.7.10.1

Cancel the common factor.

$5 x + 69 = A x + 4 A + B x^{2} + B x - 12 B + \frac{(C) {(x - 3)}^{2} (x + 4)}{x + 4}$

Step 1.7.10.2

Divide $(C) {(x - 3)}^{2}$ by $1$ .

$5 x + 69 = A x + 4 A + B x^{2} + B x - 12 B + (C) {(x - 3)}^{2}$

Step 1.7.11

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$5 x + 69 = A x + 4 A + B x^{2} + B x - 12 B + C ((x - 3) (x - 3))$

Step 1.7.12

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 1.7.12.1

Apply the distributive property.

$5 x + 69 = A x + 4 A + B x^{2} + B x - 12 B + C (x (x - 3) - 3 (x - 3))$

Step 1.7.12.2

Apply the distributive property.

$5 x + 69 = A x + 4 A + B x^{2} + B x - 12 B + C (x \cdot x + x \cdot - 3 - 3 (x - 3))$

Step 1.7.12.3

Apply the distributive property.

$5 x + 69 = A x + 4 A + B x^{2} + B x - 12 B + C (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3)$

Step 1.7.13

Simplify and combine like terms.

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Step 1.7.13.1

Simplify each term.

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Step 1.7.13.1.1

Multiply $x$ by $x$ .

$5 x + 69 = A x + 4 A + B x^{2} + B x - 12 B + C (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3)$

Step 1.7.13.1.2

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

$5 x + 69 = A x + 4 A + B x^{2} + B x - 12 B + C (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3)$

Step 1.7.13.1.3

Multiply $- 3$ by $- 3$ .

$5 x + 69 = A x + 4 A + B x^{2} + B x - 12 B + C (x^{2} - 3 x - 3 x + 9)$

Step 1.7.13.2

Subtract $3 x$ from $- 3 x$ .

$5 x + 69 = A x + 4 A + B x^{2} + B x - 12 B + C (x^{2} - 6 x + 9)$

Step 1.7.14

Apply the distributive property.

$5 x + 69 = A x + 4 A + B x^{2} + B x - 12 B + C x^{2} + C (- 6 x) + C \cdot 9$

Step 1.7.15

Simplify.

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Step 1.7.15.1

Rewrite using the commutative property of multiplication.

$5 x + 69 = A x + 4 A + B x^{2} + B x - 12 B + C x^{2} - 6 C x + C \cdot 9$

Step 1.7.15.2

Move $9$ to the left of $C$ .

$5 x + 69 = A x + 4 A + B x^{2} + B x - 12 B + C x^{2} - 6 C x + 9 \cdot C$

$5 x + 69 = A x + 4 A + B x^{2} + B x - 12 B + C x^{2} - 6 C x + 9 C$

Step 1.8

Simplify the expression.

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Step 1.8.1

Reorder $B$ and $x$ .

$5 x + 69 = A x + 4 A + B x^{2} + B x - 12 B + C x^{2} - 6 C x + 9 C$

Step 1.8.2

Move $- 12 B$ .

$5 x + 69 = A x + 4 A + B x^{2} + B x + C x^{2} - 6 C x - 12 B + 9 C$

Step 1.8.3

Move $4 A$ .

$5 x + 69 = A x + B x^{2} + B x + C x^{2} - 6 C x + 4 A - 12 B + 9 C$

Step 1.8.4

Move $B x$ .

$5 x + 69 = A x + B x^{2} + C x^{2} + B x - 6 C x + 4 A - 12 B + 9 C$

Step 1.8.5

Move $A x$ .

$5 x + 69 = B x^{2} + C x^{2} + A x + B x - 6 C x + 4 A - 12 B + 9 C$

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.

$0 = B + C$

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.

$5 = A + B - 6 C$

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.

$69 = 4 A - 12 B + 9 C$

Step 2.4

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

$0 = B + C$

$5 = A + B - 6 C$

$69 = 4 A - 12 B + 9 C$

$0 = B + C$

$5 = A + B - 6 C$

$69 = 4 A - 12 B + 9 C$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $B$ in $0 = B + C$ .

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Step 3.1.1

Rewrite the equation as $B + C = 0$ .

$B + C = 0$

$5 = A + B - 6 C$

$69 = 4 A - 12 B + 9 C$

Step 3.1.2

Subtract $C$ from both sides of the equation.

$B = - C$

$5 = A + B - 6 C$

$69 = 4 A - 12 B + 9 C$

$B = - C$

$5 = A + B - 6 C$

$69 = 4 A - 12 B + 9 C$

Step 3.2

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

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Step 3.2.1

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

$5 = A - C - 6 C$

$B = - C$

$69 = 4 A - 12 B + 9 C$

Step 3.2.2

Simplify $5 = A - C - 6 C$ .

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Step 3.2.2.1

Simplify the left side.

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Step 3.2.2.1.1

Remove parentheses.

$5 = A - C - 6 C$

$B = - C$

$69 = 4 A - 12 B + 9 C$

$5 = A - C - 6 C$

$B = - C$

$69 = 4 A - 12 B + 9 C$

Step 3.2.2.2

Simplify the right side.

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Step 3.2.2.2.1

Subtract $6 C$ from $- C$ .

$5 = A - 7 C$

$B = - C$

$69 = 4 A - 12 B + 9 C$

$5 = A - 7 C$

$B = - C$

$69 = 4 A - 12 B + 9 C$

$5 = A - 7 C$

$B = - C$

$69 = 4 A - 12 B + 9 C$

Step 3.2.3

Replace all occurrences of $B$ in $69 = 4 A - 12 B + 9 C$ with $- C$ .

$69 = 4 A - 12 (- C) + 9 C$

$5 = A - 7 C$

$B = - C$

Step 3.2.4

Simplify the right side.

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Step 3.2.4.1

Simplify $4 A - 12 (- C) + 9 C$ .

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Step 3.2.4.1.1

Multiply $- 1$ by $- 12$ .

$69 = 4 A + 12 C + 9 C$

$5 = A - 7 C$

$B = - C$

Step 3.2.4.1.2

Add $12 C$ and $9 C$ .

$69 = 4 A + 21 C$

$5 = A - 7 C$

$B = - C$

$69 = 4 A + 21 C$

$5 = A - 7 C$

$B = - C$

$69 = 4 A + 21 C$

$5 = A - 7 C$

$B = - C$

$69 = 4 A + 21 C$

$5 = A - 7 C$

$B = - C$

Step 3.3

Solve for $A$ in $5 = A - 7 C$ .

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Step 3.3.1

Rewrite the equation as $A - 7 C = 5$ .

$A - 7 C = 5$

$69 = 4 A + 21 C$

$B = - C$

Step 3.3.2

Add $7 C$ to both sides of the equation.

$A = 5 + 7 C$

$69 = 4 A + 21 C$

$B = - C$

$A = 5 + 7 C$

$69 = 4 A + 21 C$

$B = - C$

Step 3.4

Replace all occurrences of $A$ with $5 + 7 C$ in each equation.

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Step 3.4.1

Replace all occurrences of $A$ in $69 = 4 A + 21 C$ with $5 + 7 C$ .

$69 = 4 (5 + 7 C) + 21 C$

$A = 5 + 7 C$

$B = - C$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $4 (5 + 7 C) + 21 C$ .

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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.

$69 = 4 \cdot 5 + 4 (7 C) + 21 C$

$A = 5 + 7 C$

$B = - C$

Step 3.4.2.1.1.2

Multiply $4$ by $5$ .

$69 = 20 + 4 (7 C) + 21 C$

$A = 5 + 7 C$

$B = - C$

Step 3.4.2.1.1.3

Multiply $7$ by $4$ .

$69 = 20 + 28 C + 21 C$

$A = 5 + 7 C$

$B = - C$

$69 = 20 + 28 C + 21 C$

$A = 5 + 7 C$

$B = - C$

Step 3.4.2.1.2

Add $28 C$ and $21 C$ .

$69 = 20 + 49 C$

$A = 5 + 7 C$

$B = - C$

$69 = 20 + 49 C$

$A = 5 + 7 C$

$B = - C$

$69 = 20 + 49 C$

$A = 5 + 7 C$

$B = - C$

$69 = 20 + 49 C$

$A = 5 + 7 C$

$B = - C$

Step 3.5

Solve for $C$ in $69 = 20 + 49 C$ .

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Step 3.5.1

Rewrite the equation as $20 + 49 C = 69$ .

$20 + 49 C = 69$

$A = 5 + 7 C$

$B = - C$

Step 3.5.2

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

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Step 3.5.2.1

Subtract $20$ from both sides of the equation.

$49 C = 69 - 20$

$A = 5 + 7 C$

$B = - C$

Step 3.5.2.2

Subtract $20$ from $69$ .

$49 C = 49$

$A = 5 + 7 C$

$B = - C$

$49 C = 49$

$A = 5 + 7 C$

$B = - C$

Step 3.5.3

Divide each term in $49 C = 49$ by $49$ and simplify.

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Step 3.5.3.1

Divide each term in $49 C = 49$ by $49$ .

$\frac{49 C}{49} = \frac{49}{49}$

$A = 5 + 7 C$

$B = - C$

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $49$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

$\frac{49 C}{49} = \frac{49}{49}$

$A = 5 + 7 C$

$B = - C$

Step 3.5.3.2.1.2

Divide $C$ by $1$ .

$C = \frac{49}{49}$

$A = 5 + 7 C$

$B = - C$

$C = \frac{49}{49}$

$A = 5 + 7 C$

$B = - C$

$C = \frac{49}{49}$

$A = 5 + 7 C$

$B = - C$

Step 3.5.3.3

Simplify the right side.

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Step 3.5.3.3.1

Divide $49$ by $49$ .

$C = 1$

$A = 5 + 7 C$

$B = - C$

$C = 1$

$A = 5 + 7 C$

$B = - C$

$C = 1$

$A = 5 + 7 C$

$B = - C$

$C = 1$

$A = 5 + 7 C$

$B = - C$

Step 3.6

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

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Step 3.6.1

Replace all occurrences of $C$ in $A = 5 + 7 C$ with $1$ .

$A = 5 + 7 (1)$

$C = 1$

$B = - C$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Simplify $5 + 7 (1)$ .

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Step 3.6.2.1.1

Multiply $7$ by $1$ .

$A = 5 + 7$

$C = 1$

$B = - C$

Step 3.6.2.1.2

Add $5$ and $7$ .

$A = 12$

$C = 1$

$B = - C$

$A = 12$

$C = 1$

$B = - C$

$A = 12$

$C = 1$

$B = - C$

Step 3.6.3

Replace all occurrences of $C$ in $B = - C$ with $1$ .

$B = - (1)$

$A = 12$

$C = 1$

Step 3.6.4

Simplify the right side.

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Step 3.6.4.1

Multiply $- 1$ by $1$ .

$B = - 1$

$A = 12$

$C = 1$

$B = - 1$

$A = 12$

$C = 1$

$B = - 1$

$A = 12$

$C = 1$

Step 3.7

List all of the solutions.

$B = - 1, A = 12, C = 1$

Step 4

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

$\frac{12}{{(x - 3)}^{2}} + \frac{- 1}{x - 3} + \frac{1}{x + 4}$