Calculus Examples

$\frac{3 x - 1}{(x - 3) (x + 6)}$

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 $A$ .

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

Step 1.2

$\frac{A}{x - 3} + \frac{B}{x + 6}$

Step 1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(x - 3) (x + 6)$ .

$\frac{(3 x - 1) (x - 3) (x + 6)}{(x - 3) (x + 6)} = \frac{(A) (x - 3) (x + 6)}{x - 3} + \frac{(B) (x - 3) (x + 6)}{x + 6}$

Step 1.4

Cancel the common factor of $x - 3$ .

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Step 1.4.1

Cancel the common factor.

$\frac{(3 x - 1) (x - 3) (x + 6)}{(x - 3) (x + 6)} = \frac{(A) (x - 3) (x + 6)}{x - 3} + \frac{(B) (x - 3) (x + 6)}{x + 6}$

Step 1.4.2

Rewrite the expression.

$\frac{(3 x - 1) (x + 6)}{x + 6} = \frac{(A) (x - 3) (x + 6)}{x - 3} + \frac{(B) (x - 3) (x + 6)}{x + 6}$

Step 1.5

Cancel the common factor of $x + 6$ .

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Step 1.5.1

Cancel the common factor.

$\frac{(3 x - 1) (x + 6)}{x + 6} = \frac{(A) (x - 3) (x + 6)}{x - 3} + \frac{(B) (x - 3) (x + 6)}{x + 6}$

Step 1.5.2

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

$3 x - 1 = \frac{(A) (x - 3) (x + 6)}{x - 3} + \frac{(B) (x - 3) (x + 6)}{x + 6}$

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.

$3 x - 1 = \frac{A (x - 3) (x + 6)}{x - 3} + \frac{(B) (x - 3) (x + 6)}{x + 6}$

Step 1.6.1.2

Divide $(A) (x + 6)$ by $1$ .

$3 x - 1 = (A) (x + 6) + \frac{(B) (x - 3) (x + 6)}{x + 6}$

Step 1.6.2

Apply the distributive property.

$3 x - 1 = A x + A \cdot 6 + \frac{(B) (x - 3) (x + 6)}{x + 6}$

Step 1.6.3

Move $6$ to the left of $A$ .

$3 x - 1 = A x + 6 \cdot A + \frac{(B) (x - 3) (x + 6)}{x + 6}$

Step 1.6.4

Cancel the common factor of $x + 6$ .

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Step 1.6.4.1

Cancel the common factor.

$3 x - 1 = A x + 6 A + \frac{(B) (x - 3) (x + 6)}{x + 6}$

Step 1.6.4.2

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

$3 x - 1 = A x + 6 A + (B) (x - 3)$

Step 1.6.5

Apply the distributive property.

$3 x - 1 = A x + 6 A + B x + B \cdot - 3$

Step 1.6.6

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

$3 x - 1 = A x + 6 A + B x - 3 B$

Step 1.7

Move $6 A$ .

$3 x - 1 = A x + B x + 6 A - 3 B$

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.

$3 = A + B$

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.

$- 1 = 6 A - 3 B$

Step 2.3

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

$3 = A + B$

$- 1 = 6 A - 3 B$

$3 = A + B$

$- 1 = 6 A - 3 B$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $A$ in $3 = A + B$ .

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Step 3.1.1

Rewrite the equation as $A + B = 3$ .

$A + B = 3$

$- 1 = 6 A - 3 B$

Step 3.1.2

Subtract $B$ from both sides of the equation.

$A = 3 - B$

$- 1 = 6 A - 3 B$

$A = 3 - B$

$- 1 = 6 A - 3 B$

Step 3.2

Replace all occurrences of $A$ with $3 - B$ in each equation.

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Step 3.2.1

Replace all occurrences of $A$ in $- 1 = 6 A - 3 B$ with $3 - B$ .

$- 1 = 6 (3 - B) - 3 B$

$A = 3 - B$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $6 (3 - B) - 3 B$ .

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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.

$- 1 = 6 \cdot 3 + 6 (- B) - 3 B$

$A = 3 - B$

Step 3.2.2.1.1.2

Multiply $6$ by $3$ .

$- 1 = 18 + 6 (- B) - 3 B$

$A = 3 - B$

Step 3.2.2.1.1.3

Multiply $- 1$ by $6$ .

$- 1 = 18 - 6 B - 3 B$

$A = 3 - B$

$- 1 = 18 - 6 B - 3 B$

$A = 3 - B$

Step 3.2.2.1.2

Subtract $3 B$ from $- 6 B$ .

$- 1 = 18 - 9 B$

$A = 3 - B$

$- 1 = 18 - 9 B$

$A = 3 - B$

$- 1 = 18 - 9 B$

$A = 3 - B$

$- 1 = 18 - 9 B$

$A = 3 - B$

Step 3.3

Solve for $B$ in $- 1 = 18 - 9 B$ .

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Step 3.3.1

Rewrite the equation as $18 - 9 B = - 1$ .

$18 - 9 B = - 1$

$A = 3 - B$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $18$ from both sides of the equation.

$- 9 B = - 1 - 18$

$A = 3 - B$

Step 3.3.2.2

Subtract $18$ from $- 1$ .

$- 9 B = - 19$

$A = 3 - B$

$- 9 B = - 19$

$A = 3 - B$

Step 3.3.3

Divide each term in $- 9 B = - 19$ by $- 9$ and simplify.

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Step 3.3.3.1

Divide each term in $- 9 B = - 19$ by $- 9$ .

$\frac{- 9 B}{- 9} = \frac{- 19}{- 9}$

$A = 3 - B$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 9$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 9 B}{- 9} = \frac{- 19}{- 9}$

$A = 3 - B$

Step 3.3.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{- 19}{- 9}$

$A = 3 - B$

$B = \frac{- 19}{- 9}$

$A = 3 - B$

$B = \frac{- 19}{- 9}$

$A = 3 - B$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Dividing two negative values results in a positive value.

$B = \frac{19}{9}$

$A = 3 - B$

$B = \frac{19}{9}$

$A = 3 - B$

$B = \frac{19}{9}$

$A = 3 - B$

$B = \frac{19}{9}$

$A = 3 - B$

Step 3.4

Replace all occurrences of $B$ with $\frac{19}{9}$ in each equation.

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Step 3.4.1

Replace all occurrences of $B$ in $A = 3 - B$ with $\frac{19}{9}$ .

$A = 3 - (\frac{19}{9})$

$B = \frac{19}{9}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $3 - (\frac{19}{9})$ .

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Step 3.4.2.1.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$A = 3 \cdot \frac{9}{9} - \frac{19}{9}$

$B = \frac{19}{9}$

Step 3.4.2.1.2

Combine $3$ and $\frac{9}{9}$ .

$A = \frac{3 \cdot 9}{9} - \frac{19}{9}$

$B = \frac{19}{9}$

Step 3.4.2.1.3

Combine the numerators over the common denominator.

$A = \frac{3 \cdot 9 - 19}{9}$

$B = \frac{19}{9}$

Step 3.4.2.1.4

Simplify the numerator.

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Step 3.4.2.1.4.1

Multiply $3$ by $9$ .

$A = \frac{27 - 19}{9}$

$B = \frac{19}{9}$

Step 3.4.2.1.4.2

Subtract $19$ from $27$ .

$A = \frac{8}{9}$

$B = \frac{19}{9}$

$A = \frac{8}{9}$

$B = \frac{19}{9}$

$A = \frac{8}{9}$

$B = \frac{19}{9}$

$A = \frac{8}{9}$

$B = \frac{19}{9}$

$A = \frac{8}{9}$

$B = \frac{19}{9}$

Step 3.5

List all of the solutions.

$A = \frac{8}{9}, B = \frac{19}{9}$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{x - 3} + \frac{B}{x + 6}$ with the values found for $A$ and $B$ .

$\frac{\frac{8}{9}}{x - 3} + \frac{\frac{19}{9}}{x + 6}$