Precalculus Examples

\frac{- 3}{(x - 3) (x - 2)}

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

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

A

$A$ .

\frac{A}{x - 3}

$\frac{A}{x - 3}$

Step 1.2

B

$B$ .

\frac{A}{x - 3} + \frac{B}{x - 2}

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

Step 1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is

(x - 3) (x - 2)

$(x - 3) (x - 2)$ .

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

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

Step 1.4

Cancel the common factor of

x - 3

$x - 3$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

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

Step 1.5

Cancel the common factor of

$x - 2$ .

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Step 1.5.1

Cancel the common factor.

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

Step 1.5.2

Divide

$- 3$ by

$1$ .

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

Step 1.6

Simplify each term.

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Step 1.6.1

Cancel the common factor of

$x - 3$ .

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Step 1.6.1.1

Cancel the common factor.

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

Step 1.6.1.2

Divide

$(A) (x - 2)$ by

$1$ .

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

Step 1.6.2

Apply the distributive property.

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

Step 1.6.3

Move

$- 2$ to the left of

$A$ .

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

Step 1.6.4

Cancel the common factor of

$x - 2$ .

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Step 1.6.4.1

Cancel the common factor.

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

Step 1.6.4.2

Divide

$(B) (x - 3)$ by

$1$ .

$- 3 = A x - 2 A + (B) (x - 3)$

Step 1.6.5

Apply the distributive property.

$- 3 = A x - 2 A + B x + B \cdot - 3$

Step 1.6.6

Move

$- 3$ to the left of

$B$ .

$- 3 = A x - 2 A + B x - 3 B$

Step 1.7

Move

$- 2 A$ .

$- 3 = A x + B x - 2 A - 3 B$

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

$x$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = A + B$

Step 2.2

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.

$- 3 = - 2 A - 3 B$

Step 2.3

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

$0 = A + B$

$- 3 = - 2 A - 3 B$

$0 = A + B$

$- 3 = - 2 A - 3 B$

Step 3

Solve the system of equations.

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Step 3.1

Solve for

$A$ in

$0 = A + B$ .

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Step 3.1.1

Rewrite the equation as

$A + B = 0$ .

$A + B = 0$

$- 3 = - 2 A - 3 B$

Step 3.1.2

Subtract

$B$ from both sides of the equation.

$A = - B$

$- 3 = - 2 A - 3 B$

$A = - B$

$- 3 = - 2 A - 3 B$

Step 3.2

Replace all occurrences of

$A$ with

$- B$ in each equation.

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Step 3.2.1

Replace all occurrences of

$A$ in

$- 3 = - 2 A - 3 B$ with

$- B$ .

$- 3 = - 2 (- B) - 3 B$

$A = - B$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify

$- 2 (- B) - 3 B$ .

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Step 3.2.2.1.1

Multiply

$- 1$ by

$- 2$ .

$- 3 = 2 B - 3 B$

$A = - B$

Step 3.2.2.1.2

Subtract

$3 B$ from

$2 B$ .

$- 3 = - B$

$A = - B$

$- 3 = - B$

$A = - B$

$- 3 = - B$

$A = - B$

$- 3 = - B$

$A = - B$

Step 3.3

Solve for

$B$ in

$- 3 = - B$ .

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Step 3.3.1

Rewrite the equation as

$- B = - 3$ .

$- B = - 3$

$A = - B$

Step 3.3.2

Divide each term in

$- B = - 3$ by

$- 1$ and simplify.

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Step 3.3.2.1

Divide each term in

$- B = - 3$ by

$- 1$ .

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

$A = - B$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Dividing two negative values results in a positive value.

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

$A = - B$

Step 3.3.2.2.2

Divide

$B$ by

$1$ .

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

$A = - B$

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

$A = - B$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Divide

$- 3$ by

$- 1$ .

$B = 3$

$A = - B$

$B = 3$

$A = - B$

$B = 3$

$A = - B$

$B = 3$

$A = - B$

Step 3.4

Replace all occurrences of

$B$ with

$3$ in each equation.

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Step 3.4.1

Replace all occurrences of

$B$ in

$A = - B$ with

$3$ .

$A = - (3)$

$B = 3$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Multiply

$- 1$ by

$3$ .

$A = - 3$

$B = 3$

$A = - 3$

$B = 3$

$A = - 3$

$B = 3$

Step 3.5

List all of the solutions.

$A = - 3, B = 3$

Step 4

Replace each of the partial fraction coefficients in

$\frac{A}{x - 3} + \frac{B}{x - 2}$ with the values found for

$A$ and

$B$ .

$\frac{- 3}{x - 3} + \frac{3}{x - 2}$

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