Algebra Examples

$\frac{1}{x (x + 1) (x + 2)}$

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 $B$ .

$\frac{A}{x} + \frac{B}{x + 1}$

Step 1.2

$\frac{A}{x} + \frac{B}{x + 1} + \frac{C}{x + 2}$

Step 1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $x (x + 1) (x + 2)$ .

$\frac{1 (x (x + 1) (x + 2))}{x (x + 1) (x + 2)} = \frac{A (x (x + 1) (x + 2))}{x} + \frac{(B) (x (x + 1) (x + 2))}{x + 1} + \frac{(C) (x (x + 1) (x + 2))}{x + 2}$

Step 1.4

Cancel the common factor of $x$ .

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Step 1.4.1

Cancel the common factor.

$\frac{1 (x (x + 1) (x + 2))}{x (x + 1) (x + 2)} = \frac{A (x (x + 1) (x + 2))}{x} + \frac{(B) (x (x + 1) (x + 2))}{x + 1} + \frac{(C) (x (x + 1) (x + 2))}{x + 2}$

Step 1.4.2

Rewrite the expression.

$\frac{1 ((x + 1) (x + 2))}{(x + 1) (x + 2)} = \frac{A (x (x + 1) (x + 2))}{x} + \frac{(B) (x (x + 1) (x + 2))}{x + 1} + \frac{(C) (x (x + 1) (x + 2))}{x + 2}$

Step 1.5

Cancel the common factor of $x + 1$ .

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Step 1.5.1

Cancel the common factor.

$\frac{1 ((x + 1) (x + 2))}{(x + 1) (x + 2)} = \frac{A (x (x + 1) (x + 2))}{x} + \frac{(B) (x (x + 1) (x + 2))}{x + 1} + \frac{(C) (x (x + 1) (x + 2))}{x + 2}$

Step 1.5.2

Rewrite the expression.

$\frac{1 (x + 2)}{x + 2} = \frac{A (x (x + 1) (x + 2))}{x} + \frac{(B) (x (x + 1) (x + 2))}{x + 1} + \frac{(C) (x (x + 1) (x + 2))}{x + 2}$

Step 1.6

Cancel the common factor of $x + 2$ .

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Step 1.6.1

Cancel the common factor.

$\frac{1 (x + 2)}{x + 2} = \frac{A (x (x + 1) (x + 2))}{x} + \frac{(B) (x (x + 1) (x + 2))}{x + 1} + \frac{(C) (x (x + 1) (x + 2))}{x + 2}$

Step 1.6.2

Rewrite the expression.

$1 = \frac{A (x (x + 1) (x + 2))}{x} + \frac{(B) (x (x + 1) (x + 2))}{x + 1} + \frac{(C) (x (x + 1) (x + 2))}{x + 2}$

Step 1.7

Simplify each term.

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Step 1.7.1

Cancel the common factor of $x$ .

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Step 1.7.1.1

Cancel the common factor.

$1 = \frac{A (x (x + 1) (x + 2))}{x} + \frac{(B) (x (x + 1) (x + 2))}{x + 1} + \frac{(C) (x (x + 1) (x + 2))}{x + 2}$

Step 1.7.1.2

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

$1 = A ((x + 1) (x + 2)) + \frac{(B) (x (x + 1) (x + 2))}{x + 1} + \frac{(C) (x (x + 1) (x + 2))}{x + 2}$

Step 1.7.2

Expand $(x + 1) (x + 2)$ using the FOIL Method.

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Step 1.7.2.1

Apply the distributive property.

$1 = A (x (x + 2) + 1 (x + 2)) + \frac{(B) (x (x + 1) (x + 2))}{x + 1} + \frac{(C) (x (x + 1) (x + 2))}{x + 2}$

Step 1.7.2.2

Apply the distributive property.

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

Step 1.7.2.3

Apply the distributive property.

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

Step 1.7.3

Simplify and combine like terms.

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Step 1.7.3.1

Simplify each term.

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Step 1.7.3.1.1

Multiply $x$ by $x$ .

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

Step 1.7.3.1.2

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

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

Step 1.7.3.1.3

Multiply $x$ by $1$ .

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

Step 1.7.3.1.4

Multiply $2$ by $1$ .

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

Step 1.7.3.2

Add $2 x$ and $x$ .

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

Step 1.7.4

Apply the distributive property.

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

Step 1.7.5

Simplify.

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Step 1.7.5.1

Rewrite using the commutative property of multiplication.

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

Step 1.7.5.2

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

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

Step 1.7.6

Cancel the common factor of $x + 1$ .

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Step 1.7.6.1

Cancel the common factor.

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

Step 1.7.6.2

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

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

Step 1.7.7

Apply the distributive property.

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

Step 1.7.8

Multiply $x$ by $x$ .

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

Step 1.7.9

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

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

Step 1.7.10

Apply the distributive property.

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

Step 1.7.11

Rewrite using the commutative property of multiplication.

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

Step 1.7.12

Cancel the common factor of $x + 2$ .

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Step 1.7.12.1

Cancel the common factor.

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

Step 1.7.12.2

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

$1 = A x^{2} + 3 A x + 2 A + B x^{2} + 2 B x + (C) (x (x + 1))$

Step 1.7.13

Apply the distributive property.

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

Step 1.7.14

Multiply $x$ by $x$ .

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

Step 1.7.15

Multiply $x$ by $1$ .

$1 = A x^{2} + 3 A x + 2 A + B x^{2} + 2 B x + C (x^{2} + x)$

Step 1.7.16

Apply the distributive property.

$1 = A x^{2} + 3 A x + 2 A + B x^{2} + 2 B x + C x^{2} + C x$

Step 1.8

Simplify the expression.

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Step 1.8.1

Move $A$ .

$1 = A x^{2} + 3 x A + 2 A + B x^{2} + 2 B x + C x^{2} + C x$

Step 1.8.2

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

$1 = A x^{2} + 3 x A + 2 A + B x^{2} + 2 B x + C x^{2} + C x$

Step 1.8.3

Move $B$ .

$1 = A x^{2} + 3 x A + 2 A + B x^{2} + 2 x B + C x^{2} + C x$

Step 1.8.4

Move $2 A$ .

$1 = A x^{2} + 3 x A + B x^{2} + 2 x B + C x^{2} + C x + 2 A$

Step 1.8.5

Move $2 x B$ .

$1 = A x^{2} + 3 x A + B x^{2} + C x^{2} + 2 x B + C x + 2 A$

Step 1.8.6

Move $3 x A$ .

$1 = A x^{2} + B x^{2} + C x^{2} + 3 x A + 2 x B + C x + 2 A$

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.

$0 = A + B + C$

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.

$0 = 3 A + 2 B + C$

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.

$1 = 2 A$

Step 2.4

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

$0 = A + B + C$

$0 = 3 A + 2 B + C$

$1 = 2 A$

$0 = A + B + C$

$0 = 3 A + 2 B + C$

$1 = 2 A$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $A$ in $1 = 2 A$ .

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Step 3.1.1

Rewrite the equation as $2 A = 1$ .

$2 A = 1$

$0 = A + B + C$

$0 = 3 A + 2 B + C$

Step 3.1.2

Divide each term in $2 A = 1$ by $2$ and simplify.

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Step 3.1.2.1

Divide each term in $2 A = 1$ by $2$ .

$\frac{2 A}{2} = \frac{1}{2}$

$0 = A + B + C$

$0 = 3 A + 2 B + C$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $2$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{2 A}{2} = \frac{1}{2}$

$0 = A + B + C$

$0 = 3 A + 2 B + C$

Step 3.1.2.2.1.2

Divide $A$ by $1$ .

$A = \frac{1}{2}$

$0 = A + B + C$

$0 = 3 A + 2 B + C$

$A = \frac{1}{2}$

$0 = A + B + C$

$0 = 3 A + 2 B + C$

$A = \frac{1}{2}$

$0 = A + B + C$

$0 = 3 A + 2 B + C$

$A = \frac{1}{2}$

$0 = A + B + C$

$0 = 3 A + 2 B + C$

$A = \frac{1}{2}$

$0 = A + B + C$

$0 = 3 A + 2 B + C$

Step 3.2

Replace all occurrences of $A$ with $\frac{1}{2}$ in each equation.

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Step 3.2.1

Replace all occurrences of $A$ in $0 = A + B + C$ with $\frac{1}{2}$ .

$0 = (\frac{1}{2}) + B + C$

$A = \frac{1}{2}$

$0 = 3 A + 2 B + C$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Remove parentheses.

$0 = \frac{1}{2} + B + C$

$A = \frac{1}{2}$

$0 = 3 A + 2 B + C$

$0 = \frac{1}{2} + B + C$

$A = \frac{1}{2}$

$0 = 3 A + 2 B + C$

Step 3.2.3

Replace all occurrences of $A$ in $0 = 3 A + 2 B + C$ with $\frac{1}{2}$ .

$0 = 3 (\frac{1}{2}) + 2 B + C$

$0 = \frac{1}{2} + B + C$

$A = \frac{1}{2}$

Step 3.2.4

Simplify the right side.

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Step 3.2.4.1

Combine $3$ and $\frac{1}{2}$ .

$0 = \frac{3}{2} + 2 B + C$

$0 = \frac{1}{2} + B + C$

$A = \frac{1}{2}$

$0 = \frac{3}{2} + 2 B + C$

$0 = \frac{1}{2} + B + C$

$A = \frac{1}{2}$

$0 = \frac{3}{2} + 2 B + C$

$0 = \frac{1}{2} + B + C$

$A = \frac{1}{2}$

Step 3.3

Solve for $C$ in $0 = \frac{3}{2} + 2 B + C$ .

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Step 3.3.1

Rewrite the equation as $\frac{3}{2} + 2 B + C = 0$ .

$\frac{3}{2} + 2 B + C = 0$

$0 = \frac{1}{2} + B + C$

$A = \frac{1}{2}$

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 $\frac{3}{2}$ from both sides of the equation.

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

$0 = \frac{1}{2} + B + C$

$A = \frac{1}{2}$

Step 3.3.2.2

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

$C = - \frac{3}{2} - 2 B$

$0 = \frac{1}{2} + B + C$

$A = \frac{1}{2}$

$C = - \frac{3}{2} - 2 B$

$0 = \frac{1}{2} + B + C$

$A = \frac{1}{2}$

$C = - \frac{3}{2} - 2 B$

$0 = \frac{1}{2} + B + C$

$A = \frac{1}{2}$

Step 3.4

Replace all occurrences of $C$ with $- \frac{3}{2} - 2 B$ in each equation.

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Step 3.4.1

Replace all occurrences of $C$ in $0 = \frac{1}{2} + B + C$ with $- \frac{3}{2} - 2 B$ .

$0 = \frac{1}{2} + B - \frac{3}{2} - 2 B$

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

Step 3.4.2

Simplify $0 = \frac{1}{2} + B - \frac{3}{2} - 2 B$ .

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Step 3.4.2.1

Simplify the left side.

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Step 3.4.2.1.1

Remove parentheses.

$0 = \frac{1}{2} + B - \frac{3}{2} - 2 B$

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

$0 = \frac{1}{2} + B - \frac{3}{2} - 2 B$

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

Step 3.4.2.2

Simplify the right side.

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Step 3.4.2.2.1

Simplify $\frac{1}{2} + B - \frac{3}{2} - 2 B$ .

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Step 3.4.2.2.1.1

Combine the numerators over the common denominator.

$0 = B - 2 B + \frac{1 - 3}{2}$

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

Step 3.4.2.2.1.2

Simplify the expression.

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Step 3.4.2.2.1.2.1

Subtract $3$ from $1$ .

$0 = B - 2 B + \frac{- 2}{2}$

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

Step 3.4.2.2.1.2.2

Divide $- 2$ by $2$ .

$0 = B - 2 B - 1$

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

$0 = B - 2 B - 1$

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

Step 3.4.2.2.1.3

Subtract $2 B$ from $B$ .

$0 = - B - 1$

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

$0 = - B - 1$

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

$0 = - B - 1$

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

$0 = - B - 1$

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

$0 = - B - 1$

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

Step 3.5

Solve for $B$ in $0 = - B - 1$ .

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Step 3.5.1

Rewrite the equation as $- B - 1 = 0$ .

$- B - 1 = 0$

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

Step 3.5.2

Add $1$ to both sides of the equation.

$- B = 1$

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

Step 3.5.3

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

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Step 3.5.3.1

Divide each term in $- B = 1$ by $- 1$ .

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

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Dividing two negative values results in a positive value.

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

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

Step 3.5.3.2.2

Divide $B$ by $1$ .

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

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

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

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

Step 3.5.3.3

Simplify the right side.

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Step 3.5.3.3.1

Divide $1$ by $- 1$ .

$B = - 1$

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

$B = - 1$

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

$B = - 1$

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

$B = - 1$

$C = - \frac{3}{2} - 2 B$

$A = \frac{1}{2}$

Step 3.6

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

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Step 3.6.1

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

$C = - \frac{3}{2} - 2 \cdot - 1$

$B = - 1$

$A = \frac{1}{2}$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Simplify $- \frac{3}{2} - 2 \cdot - 1$ .

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Step 3.6.2.1.1

Multiply $- 2$ by $- 1$ .

$C = - \frac{3}{2} + 2$

$B = - 1$

$A = \frac{1}{2}$

Step 3.6.2.1.2

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

$C = - \frac{3}{2} + 2 \cdot \frac{2}{2}$

$B = - 1$

$A = \frac{1}{2}$

Step 3.6.2.1.3

Combine $2$ and $\frac{2}{2}$ .

$C = - \frac{3}{2} + \frac{2 \cdot 2}{2}$

$B = - 1$

$A = \frac{1}{2}$

Step 3.6.2.1.4

Combine the numerators over the common denominator.

$C = \frac{- 3 + 2 \cdot 2}{2}$

$B = - 1$

$A = \frac{1}{2}$

Step 3.6.2.1.5

Simplify the numerator.

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Step 3.6.2.1.5.1

Multiply $2$ by $2$ .

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

$B = - 1$

$A = \frac{1}{2}$

Step 3.6.2.1.5.2

Add $- 3$ and $4$ .

$C = \frac{1}{2}$

$B = - 1$

$A = \frac{1}{2}$

$C = \frac{1}{2}$

$B = - 1$

$A = \frac{1}{2}$

$C = \frac{1}{2}$

$B = - 1$

$A = \frac{1}{2}$

$C = \frac{1}{2}$

$B = - 1$

$A = \frac{1}{2}$

$C = \frac{1}{2}$

$B = - 1$

$A = \frac{1}{2}$

Step 3.7

List all of the solutions.

$C = \frac{1}{2}, B = - 1, A = \frac{1}{2}$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x} + \frac{B}{x + 1} + \frac{C}{x + 2}$ with the values found for $A$ , $B$ , and $C$ .

$\frac{\frac{1}{2}}{x} + \frac{- 1}{x + 1} + \frac{\frac{1}{2}}{x + 2}$