Trigonometry Examples

\frac{2 x - 9}{x (x + 3)}

$\frac{2 x - 9}{x (x + 3)}$

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

$B$ .

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

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

Step 1.2

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

x (x + 3)

$x (x + 3)$ .

\frac{(2 x - 9) (x (x + 3))}{x (x + 3)} = \frac{A (x (x + 3))}{x} + \frac{(B) (x (x + 3))}{x + 3}

$\frac{(2 x - 9) (x (x + 3))}{x (x + 3)} = \frac{A (x (x + 3))}{x} + \frac{(B) (x (x + 3))}{x + 3}$

Step 1.3

Cancel the common factor of

x

$x$ .

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Step 1.3.1

Cancel the common factor.

$\frac{(2 x - 9) (x (x + 3))}{x (x + 3)} = \frac{A (x (x + 3))}{x} + \frac{(B) (x (x + 3))}{x + 3}$

Step 1.3.2

Rewrite the expression.

$\frac{(2 x - 9) (x + 3)}{x + 3} = \frac{A (x (x + 3))}{x} + \frac{(B) (x (x + 3))}{x + 3}$

Step 1.4

Cancel the common factor of

$x + 3$ .

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Step 1.4.1

Cancel the common factor.

$\frac{(2 x - 9) (x + 3)}{x + 3} = \frac{A (x (x + 3))}{x} + \frac{(B) (x (x + 3))}{x + 3}$

Step 1.4.2

Divide

$2 x - 9$ by

$1$ .

$2 x - 9 = \frac{A (x (x + 3))}{x} + \frac{(B) (x (x + 3))}{x + 3}$

Step 1.5

Simplify each term.

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Step 1.5.1

Cancel the common factor of

$x$ .

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Step 1.5.1.1

Cancel the common factor.

$2 x - 9 = \frac{A (x (x + 3))}{x} + \frac{(B) (x (x + 3))}{x + 3}$

Step 1.5.1.2

Divide

$A (x + 3)$ by

$1$ .

$2 x - 9 = A (x + 3) + \frac{(B) (x (x + 3))}{x + 3}$

Step 1.5.2

Apply the distributive property.

$2 x - 9 = A x + A \cdot 3 + \frac{(B) (x (x + 3))}{x + 3}$

Step 1.5.3

Move

$3$ to the left of

$A$ .

$2 x - 9 = A x + 3 \cdot A + \frac{(B) (x (x + 3))}{x + 3}$

Step 1.5.4

Cancel the common factor of

$x + 3$ .

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Step 1.5.4.1

Cancel the common factor.

$2 x - 9 = A x + 3 A + \frac{B (x (x + 3))}{x + 3}$

Step 1.5.4.2

Divide

$(B) (x)$ by

$1$ .

$2 x - 9 = A x + 3 A + (B) (x)$

$2 x - 9 = A x + 3 A + B x$

Step 1.6

Move

$3 A$ .

$2 x - 9 = A x + B x + 3 A$

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

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

$2 = A + B$

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.

$- 9 = 3 A$

Step 2.3

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

$2 = A + B$

$- 9 = 3 A$

$2 = A + B$

$- 9 = 3 A$

Step 3

Solve the system of equations.

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Step 3.1

Solve for

$A$ in

$- 9 = 3 A$ .

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Step 3.1.1

Rewrite the equation as

$3 A = - 9$ .

$3 A = - 9$

$2 = A + B$

Step 3.1.2

Divide each term in

$3 A = - 9$ by

$3$ and simplify.

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Step 3.1.2.1

Divide each term in

$3 A = - 9$ by

$3$ .

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

$2 = A + B$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of

$3$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

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

$2 = A + B$

Step 3.1.2.2.1.2

Divide

$A$ by

$1$ .

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

$2 = A + B$

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

$2 = A + B$

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

$2 = A + B$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide

$- 9$ by

$3$ .

$A = - 3$

$2 = A + B$

$A = - 3$

$2 = A + B$

$A = - 3$

$2 = A + B$

$A = - 3$

$2 = A + B$

Step 3.2

Replace all occurrences of

$A$ with

$- 3$ in each equation.

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Step 3.2.1

Replace all occurrences of

$A$ in

$2 = A + B$ with

$- 3$ .

$2 = (- 3) + B$

$A = - 3$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Remove parentheses.

$2 = - 3 + B$

$A = - 3$

$2 = - 3 + B$

$A = - 3$

$2 = - 3 + B$

$A = - 3$

Step 3.3

Solve for

$B$ in

$2 = - 3 + B$ .

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Step 3.3.1

Rewrite the equation as

$- 3 + B = 2$ .

$- 3 + B = 2$

$A = - 3$

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

Add

$3$ to both sides of the equation.

$B = 2 + 3$

$A = - 3$

Step 3.3.2.2

Add

$2$ and

$3$ .

$B = 5$

$A = - 3$

$B = 5$

$A = - 3$

$B = 5$

$A = - 3$

Step 3.4

Solve the system of equations.

$B = 5 A = - 3$

Step 3.5

List all of the solutions.

$B = 5, A = - 3$

Step 4

Replace each of the partial fraction coefficients in

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

$A$ and

$B$ .

$- \frac{3}{x} + \frac{5}{x + 3}$