Precalculus Examples

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

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 $B$ .

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

Step 1.2

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

Step 1.3

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

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

Step 1.4

Cancel the common factor of $x$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

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

Step 1.5

Cancel the common factor of ${(x - 4)}^{2}$ .

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Step 1.5.1

Cancel the common factor.

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

Step 1.5.2

Divide $3 x - 1$ by $1$ .

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

Step 1.6

Simplify each term.

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Step 1.6.1

Cancel the common factor of $x$ .

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Step 1.6.1.1

Cancel the common factor.

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

Step 1.6.1.2

Divide $A ({(x - 4)}^{2})$ by $1$ .

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

Step 1.6.2

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

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

Step 1.6.3

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

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Step 1.6.3.1

Apply the distributive property.

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

Step 1.6.3.2

Apply the distributive property.

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

Step 1.6.3.3

Apply the distributive property.

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

Step 1.6.4

Simplify and combine like terms.

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Step 1.6.4.1

Simplify each term.

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Step 1.6.4.1.1

Multiply $x$ by $x$ .

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

Step 1.6.4.1.2

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

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

Step 1.6.4.1.3

Multiply $- 4$ by $- 4$ .

$3 x - 1 = A (x^{2} - 4 x - 4 x + 16) + \frac{(B) (x {(x - 4)}^{2})}{{(x - 4)}^{2}} + \frac{(C) (x {(x - 4)}^{2})}{x - 4}$

Step 1.6.4.2

Subtract $4 x$ from $- 4 x$ .

$3 x - 1 = A (x^{2} - 8 x + 16) + \frac{(B) (x {(x - 4)}^{2})}{{(x - 4)}^{2}} + \frac{(C) (x {(x - 4)}^{2})}{x - 4}$

Step 1.6.5

Apply the distributive property.

$3 x - 1 = A x^{2} + A (- 8 x) + A \cdot 16 + \frac{(B) (x {(x - 4)}^{2})}{{(x - 4)}^{2}} + \frac{(C) (x {(x - 4)}^{2})}{x - 4}$

Step 1.6.6

Simplify.

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Step 1.6.6.1

Rewrite using the commutative property of multiplication.

$3 x - 1 = A x^{2} - 8 A x + A \cdot 16 + \frac{(B) (x {(x - 4)}^{2})}{{(x - 4)}^{2}} + \frac{(C) (x {(x - 4)}^{2})}{x - 4}$

Step 1.6.6.2

Move $16$ to the left of $A$ .

$3 x - 1 = A x^{2} - 8 A x + 16 \cdot A + \frac{(B) (x {(x - 4)}^{2})}{{(x - 4)}^{2}} + \frac{(C) (x {(x - 4)}^{2})}{x - 4}$

Step 1.6.7

Cancel the common factor of ${(x - 4)}^{2}$ .

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Step 1.6.7.1

Cancel the common factor.

$3 x - 1 = A x^{2} - 8 A x + 16 A + \frac{B (x {(x - 4)}^{2})}{{(x - 4)}^{2}} + \frac{(C) (x {(x - 4)}^{2})}{x - 4}$

Step 1.6.7.2

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

$3 x - 1 = A x^{2} - 8 A x + 16 A + (B) (x) + \frac{(C) (x {(x - 4)}^{2})}{x - 4}$

Step 1.6.8

Cancel the common factor of ${(x - 4)}^{2}$ and $x - 4$ .

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Step 1.6.8.1

Factor $x - 4$ out of $(C) (x {(x - 4)}^{2})$ .

$3 x - 1 = A x^{2} - 8 A x + 16 A + B x + \frac{(x - 4) (C (x (x - 4)))}{x - 4}$

Step 1.6.8.2

Cancel the common factors.

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Step 1.6.8.2.1

Multiply by $1$ .

$3 x - 1 = A x^{2} - 8 A x + 16 A + B x + \frac{(x - 4) (C (x (x - 4)))}{(x - 4) \cdot 1}$

Step 1.6.8.2.2

Cancel the common factor.

$3 x - 1 = A x^{2} - 8 A x + 16 A + B x + \frac{(x - 4) (C (x (x - 4)))}{(x - 4) \cdot 1}$

Step 1.6.8.2.3

Rewrite the expression.

$3 x - 1 = A x^{2} - 8 A x + 16 A + B x + \frac{C (x (x - 4))}{1}$

Step 1.6.8.2.4

Divide $C (x (x - 4))$ by $1$ .

$3 x - 1 = A x^{2} - 8 A x + 16 A + B x + C (x (x - 4))$

Step 1.6.9

Apply the distributive property.

$3 x - 1 = A x^{2} - 8 A x + 16 A + B x + C (x \cdot x + x \cdot - 4)$

Step 1.6.10

Multiply $x$ by $x$ .

$3 x - 1 = A x^{2} - 8 A x + 16 A + B x + C (x^{2} + x \cdot - 4)$

Step 1.6.11

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

$3 x - 1 = A x^{2} - 8 A x + 16 A + B x + C (x^{2} - 4 \cdot x)$

Step 1.6.12

Apply the distributive property.

$3 x - 1 = A x^{2} - 8 A x + 16 A + B x + C x^{2} + C (- 4 x)$

Step 1.6.13

Rewrite using the commutative property of multiplication.

$3 x - 1 = A x^{2} - 8 A x + 16 A + B x + C x^{2} - 4 C x$

Step 1.7

Simplify the expression.

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Step 1.7.1

Move $A$ .

$3 x - 1 = A x^{2} - 8 x A + 16 A + B x + C x^{2} - 4 C x$

Step 1.7.2

Reorder $B$ and $x$ .

$3 x - 1 = A x^{2} - 8 x A + 16 A + B x + C x^{2} - 4 C x$

Step 1.7.3

Move $16 A$ .

$3 x - 1 = A x^{2} - 8 x A + B x + C x^{2} - 4 C x + 16 A$

Step 1.7.4

Move $B x$ .

$3 x - 1 = A x^{2} - 8 x A + C x^{2} + B x - 4 C x + 16 A$

Step 1.7.5

Move $- 8 x A$ .

$3 x - 1 = A x^{2} + C x^{2} - 8 x A + B x - 4 C x + 16 A$

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.

$0 = A + C$

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.

$3 = - 8 A + B - 4 C$

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.

$- 1 = 16 A$

Step 2.4

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

$0 = A + C$

$3 = - 8 A + B - 4 C$

$- 1 = 16 A$

$0 = A + C$

$3 = - 8 A + B - 4 C$

$- 1 = 16 A$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $A$ in $- 1 = 16 A$ .

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Step 3.1.1

Rewrite the equation as $16 A = - 1$ .

$16 A = - 1$

$0 = A + C$

$3 = - 8 A + B - 4 C$

Step 3.1.2

Divide each term in $16 A = - 1$ by $16$ and simplify.

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Step 3.1.2.1

Divide each term in $16 A = - 1$ by $16$ .

$\frac{16 A}{16} = \frac{- 1}{16}$

$0 = A + C$

$3 = - 8 A + B - 4 C$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $16$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{16 A}{16} = \frac{- 1}{16}$

$0 = A + C$

$3 = - 8 A + B - 4 C$

Step 3.1.2.2.1.2

Divide $A$ by $1$ .

$A = \frac{- 1}{16}$

$0 = A + C$

$3 = - 8 A + B - 4 C$

$A = \frac{- 1}{16}$

$0 = A + C$

$3 = - 8 A + B - 4 C$

$A = \frac{- 1}{16}$

$0 = A + C$

$3 = - 8 A + B - 4 C$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Move the negative in front of the fraction.

$A = - \frac{1}{16}$

$0 = A + C$

$3 = - 8 A + B - 4 C$

$A = - \frac{1}{16}$

$0 = A + C$

$3 = - 8 A + B - 4 C$

$A = - \frac{1}{16}$

$0 = A + C$

$3 = - 8 A + B - 4 C$

$A = - \frac{1}{16}$

$0 = A + C$

$3 = - 8 A + B - 4 C$

Step 3.2

Replace all occurrences of $A$ with $- \frac{1}{16}$ in each equation.

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Step 3.2.1

Replace all occurrences of $A$ in $0 = A + C$ with $- \frac{1}{16}$ .

$0 = (- \frac{1}{16}) + C$

$A = - \frac{1}{16}$

$3 = - 8 A + B - 4 C$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Remove parentheses.

$0 = - \frac{1}{16} + C$

$A = - \frac{1}{16}$

$3 = - 8 A + B - 4 C$

$0 = - \frac{1}{16} + C$

$A = - \frac{1}{16}$

$3 = - 8 A + B - 4 C$

Step 3.2.3

Replace all occurrences of $A$ in $3 = - 8 A + B - 4 C$ with $- \frac{1}{16}$ .

$3 = - 8 (- \frac{1}{16}) + B - 4 C$

$0 = - \frac{1}{16} + C$

$A = - \frac{1}{16}$

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

Cancel the common factor of $8$ .

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Step 3.2.4.1.1.1

Move the leading negative in $- \frac{1}{16}$ into the numerator.

$3 = - 8 (\frac{- 1}{16}) + B - 4 C$

$0 = - \frac{1}{16} + C$

$A = - \frac{1}{16}$

Step 3.2.4.1.1.2

Factor $8$ out of $- 8$ .

$3 = 8 (- 1) (\frac{- 1}{16}) + B - 4 C$

$0 = - \frac{1}{16} + C$

$A = - \frac{1}{16}$

Step 3.2.4.1.1.3

Factor $8$ out of $16$ .

$3 = 8 \cdot (- 1 \frac{- 1}{8 \cdot 2}) + B - 4 C$

$0 = - \frac{1}{16} + C$

$A = - \frac{1}{16}$

Step 3.2.4.1.1.4

Cancel the common factor.

$3 = 8 \cdot (- 1 \frac{- 1}{8 \cdot 2}) + B - 4 C$

$0 = - \frac{1}{16} + C$

$A = - \frac{1}{16}$

Step 3.2.4.1.1.5

Rewrite the expression.

$3 = - 1 (\frac{- 1}{2}) + B - 4 C$

$0 = - \frac{1}{16} + C$

$A = - \frac{1}{16}$

$3 = - 1 (\frac{- 1}{2}) + B - 4 C$

$0 = - \frac{1}{16} + C$

$A = - \frac{1}{16}$

Step 3.2.4.1.2

Move the negative in front of the fraction.

$3 = - 1 (- \frac{1}{2}) + B - 4 C$

$0 = - \frac{1}{16} + C$

$A = - \frac{1}{16}$

Step 3.2.4.1.3

Multiply $- 1 (- \frac{1}{2})$ .

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Step 3.2.4.1.3.1

Multiply $- 1$ by $- 1$ .

$3 = 1 (\frac{1}{2}) + B - 4 C$

$0 = - \frac{1}{16} + C$

$A = - \frac{1}{16}$

Step 3.2.4.1.3.2

Multiply $\frac{1}{2}$ by $1$ .

$3 = \frac{1}{2} + B - 4 C$

$0 = - \frac{1}{16} + C$

$A = - \frac{1}{16}$

$3 = \frac{1}{2} + B - 4 C$

$0 = - \frac{1}{16} + C$

$A = - \frac{1}{16}$

$3 = \frac{1}{2} + B - 4 C$

$0 = - \frac{1}{16} + C$

$A = - \frac{1}{16}$

$3 = \frac{1}{2} + B - 4 C$

$0 = - \frac{1}{16} + C$

$A = - \frac{1}{16}$

$3 = \frac{1}{2} + B - 4 C$

$0 = - \frac{1}{16} + C$

$A = - \frac{1}{16}$

Step 3.3

Solve for $C$ in $0 = - \frac{1}{16} + C$ .

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Step 3.3.1

Rewrite the equation as $- \frac{1}{16} + C = 0$ .

$- \frac{1}{16} + C = 0$

$3 = \frac{1}{2} + B - 4 C$

$A = - \frac{1}{16}$

Step 3.3.2

Add $\frac{1}{16}$ to both sides of the equation.

$C = \frac{1}{16}$

$3 = \frac{1}{2} + B - 4 C$

$A = - \frac{1}{16}$

$C = \frac{1}{16}$

$3 = \frac{1}{2} + B - 4 C$

$A = - \frac{1}{16}$

Step 3.4

Replace all occurrences of $C$ with $\frac{1}{16}$ in each equation.

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Step 3.4.1

Replace all occurrences of $C$ in $3 = \frac{1}{2} + B - 4 C$ with $\frac{1}{16}$ .

$3 = \frac{1}{2} + B - 4 (\frac{1}{16})$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $\frac{1}{2} + B - 4 (\frac{1}{16})$ .

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Step 3.4.2.1.1

Simplify each term.

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Step 3.4.2.1.1.1

Cancel the common factor of $4$ .

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Step 3.4.2.1.1.1.1

Factor $4$ out of $- 4$ .

$3 = \frac{1}{2} + B + 4 (- 1) (\frac{1}{16})$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

Step 3.4.2.1.1.1.2

Factor $4$ out of $16$ .

$3 = \frac{1}{2} + B + 4 \cdot (- 1 \frac{1}{4 \cdot 4})$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

Step 3.4.2.1.1.1.3

Cancel the common factor.

$3 = \frac{1}{2} + B + 4 \cdot (- 1 \frac{1}{4 \cdot 4})$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

Step 3.4.2.1.1.1.4

Rewrite the expression.

$3 = \frac{1}{2} + B - 1 (\frac{1}{4})$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

$3 = \frac{1}{2} + B - 1 (\frac{1}{4})$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

Step 3.4.2.1.1.2

Rewrite $- 1 (\frac{1}{4})$ as $- (\frac{1}{4})$ .

$3 = \frac{1}{2} + B - \frac{1}{4}$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

$3 = \frac{1}{2} + B - \frac{1}{4}$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

Step 3.4.2.1.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 = B + \frac{1}{2} \cdot \frac{2}{2} - \frac{1}{4}$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

Step 3.4.2.1.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.2.1.3.1

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$3 = B + \frac{2}{2 \cdot 2} - \frac{1}{4}$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

Step 3.4.2.1.3.2

Multiply $2$ by $2$ .

$3 = B + \frac{2}{4} - \frac{1}{4}$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

$3 = B + \frac{2}{4} - \frac{1}{4}$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

Step 3.4.2.1.4

Combine the numerators over the common denominator.

$3 = B + \frac{2 - 1}{4}$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

Step 3.4.2.1.5

Subtract $1$ from $2$ .

$3 = B + \frac{1}{4}$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

$3 = B + \frac{1}{4}$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

$3 = B + \frac{1}{4}$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

$3 = B + \frac{1}{4}$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

Step 3.5

Solve for $B$ in $3 = B + \frac{1}{4}$ .

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Step 3.5.1

Rewrite the equation as $B + \frac{1}{4} = 3$ .

$B + \frac{1}{4} = 3$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

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 $\frac{1}{4}$ from both sides of the equation.

$B = 3 - \frac{1}{4}$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

Step 3.5.2.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$B = 3 \cdot \frac{4}{4} - \frac{1}{4}$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

Step 3.5.2.3

Combine $3$ and $\frac{4}{4}$ .

$B = \frac{3 \cdot 4}{4} - \frac{1}{4}$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

Step 3.5.2.4

Combine the numerators over the common denominator.

$B = \frac{3 \cdot 4 - 1}{4}$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

Step 3.5.2.5

Simplify the numerator.

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Step 3.5.2.5.1

Multiply $3$ by $4$ .

$B = \frac{12 - 1}{4}$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

Step 3.5.2.5.2

Subtract $1$ from $12$ .

$B = \frac{11}{4}$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

$B = \frac{11}{4}$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

$B = \frac{11}{4}$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

$B = \frac{11}{4}$

$C = \frac{1}{16}$

$A = - \frac{1}{16}$

Step 3.6

Solve the system of equations.

$B = \frac{11}{4} C = \frac{1}{16} A = - \frac{1}{16}$

Step 3.7

List all of the solutions.

$B = \frac{11}{4}, C = \frac{1}{16}, A = - \frac{1}{16}$

Step 4

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

$- \frac{\frac{1}{16}}{x} + \frac{\frac{11}{4}}{{(x - 4)}^{2}} + \frac{\frac{1}{16}}{x - 4}$