Precalculus Examples

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

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 $A$ .

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

Step 1.2

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

Step 1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is ${(x - 3)}^{2}$ .

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

Step 1.4

Cancel the common factor of ${(x - 3)}^{2}$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Divide $3 x$ by $1$ .

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

Step 1.5

Simplify each term.

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Step 1.5.1

Cancel the common factor of ${(x - 3)}^{2}$ .

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Step 1.5.1.1

Cancel the common factor.

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

Step 1.5.1.2

Divide $A$ by $1$ .

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

Step 1.5.2

Cancel the common factor of ${(x - 3)}^{2}$ and $x - 3$ .

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Step 1.5.2.1

Factor $x - 3$ out of $(B) {(x - 3)}^{2}$ .

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

Step 1.5.2.2

Cancel the common factors.

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Step 1.5.2.2.1

Multiply by $1$ .

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

Step 1.5.2.2.2

Cancel the common factor.

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

Step 1.5.2.2.3

Rewrite the expression.

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

Step 1.5.2.2.4

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

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

Step 1.5.3

Apply the distributive property.

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

Step 1.5.4

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

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

Step 1.6

Reorder $A$ and $B x$ .

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

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.

$3 = B$

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.

$0 = A - 3 B$

Step 2.3

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

$3 = B$

$0 = A - 3 B$

$3 = B$

$0 = A - 3 B$

Step 3

Solve the system of equations.

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Step 3.1

Rewrite the equation as $B = 3$ .

$B = 3$

$0 = A - 3 B$

Step 3.2

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

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Step 3.2.1

Replace all occurrences of $B$ in $0 = A - 3 B$ with $3$ .

$0 = A - 3 \cdot 3$

$B = 3$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- 3$ by $3$ .

$0 = A - 9$

$B = 3$

$0 = A - 9$

$B = 3$

$0 = A - 9$

$B = 3$

Step 3.3

Solve for $A$ in $0 = A - 9$ .

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Step 3.3.1

Rewrite the equation as $A - 9 = 0$ .

$A - 9 = 0$

$B = 3$

Step 3.3.2

Add $9$ to both sides of the equation.

$A = 9$

$B = 3$

$A = 9$

$B = 3$

Step 3.4

Solve the system of equations.

$A = 9 B = 3$

Step 3.5

List all of the solutions.

$A = 9, B = 3$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{{(x - 3)}^{2}} + \frac{B}{x - 3}$ with the values found for $A$ and $B$ .

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