Precalculus Examples

$\frac{10 x + 20}{x^{3} - 2 x^{2} - 4 x + 8}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

Factor the fraction.

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Step 1.1.1

Factor $10$ out of $10 x + 20$ .

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Step 1.1.1.1

Factor $10$ out of $10 x$ .

$\frac{10 (x) + 20}{x^{3} - 2 x^{2} - 4 x + 8}$

Step 1.1.1.2

Factor $10$ out of $20$ .

$\frac{10 x + 10 \cdot 2}{x^{3} - 2 x^{2} - 4 x + 8}$

Step 1.1.1.3

Factor $10$ out of $10 x + 10 \cdot 2$ .

$\frac{10 (x + 2)}{x^{3} - 2 x^{2} - 4 x + 8}$

Step 1.1.2

Factor out the greatest common factor from each group.

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Step 1.1.2.1

Group the first two terms and the last two terms.

$\frac{10 (x + 2)}{(x^{3} - 2 x^{2}) - 4 x + 8}$

Step 1.1.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{10 (x + 2)}{x^{2} (x - 2) - 4 (x - 2)}$

Step 1.1.3

Factor the polynomial by factoring out the greatest common factor, $x - 2$ .

$\frac{10 (x + 2)}{(x - 2) (x^{2} - 4)}$

Step 1.1.4

Rewrite $4$ as $2^{2}$ .

$\frac{10 (x + 2)}{(x - 2) (x^{2} - 2^{2})}$

Step 1.1.5

Factor.

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Step 1.1.5.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\frac{10 (x + 2)}{(x - 2) ((x + 2) (x - 2))}$

Step 1.1.5.2

Remove unnecessary parentheses.

$\frac{10 (x + 2)}{(x - 2) (x + 2) (x - 2)}$

Step 1.1.6

Combine exponents.

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Step 1.1.6.1

Raise $x - 2$ to the power of $1$ .

$\frac{10 (x + 2)}{{(x - 2)}^{1} (x - 2) (x + 2)}$

Step 1.1.6.2

Raise $x - 2$ to the power of $1$ .

$\frac{10 (x + 2)}{{(x - 2)}^{1} {(x - 2)}^{1} (x + 2)}$

Step 1.1.6.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{10 (x + 2)}{{(x - 2)}^{1 + 1} (x + 2)}$

Step 1.1.6.4

Add $1$ and $1$ .

$\frac{10 (x + 2)}{{(x - 2)}^{2} (x + 2)}$

Step 1.1.7

Reduce the expression $\frac{10 (x + 2)}{{(x - 2)}^{2} (x + 2)}$ by cancelling the common factors.

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Step 1.1.7.1

Cancel the common factor.

$\frac{10 (x + 2)}{{(x - 2)}^{2} (x + 2)}$

Step 1.1.7.2

Rewrite the expression.

$\frac{10}{{(x - 2)}^{2}}$

Step 1.2

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 $A$ .

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

Step 1.3

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

Step 1.4

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

$\frac{10 {(x - 2)}^{2}}{{(x - 2)}^{2}} = \frac{(A) {(x - 2)}^{2}}{{(x - 2)}^{2}} + \frac{(B) {(x - 2)}^{2}}{x - 2}$

Step 1.5

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

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Step 1.5.1

Cancel the common factor.

$\frac{10 {(x - 2)}^{2}}{{(x - 2)}^{2}} = \frac{(A) {(x - 2)}^{2}}{{(x - 2)}^{2}} + \frac{(B) {(x - 2)}^{2}}{x - 2}$

Step 1.5.2

Divide $10$ by $1$ .

$10 = \frac{(A) {(x - 2)}^{2}}{{(x - 2)}^{2}} + \frac{(B) {(x - 2)}^{2}}{x - 2}$

Step 1.6

Simplify each term.

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Step 1.6.1

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

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Step 1.6.1.1

Cancel the common factor.

$10 = \frac{A {(x - 2)}^{2}}{{(x - 2)}^{2}} + \frac{(B) {(x - 2)}^{2}}{x - 2}$

Step 1.6.1.2

Divide $A$ by $1$ .

$10 = A + \frac{(B) {(x - 2)}^{2}}{x - 2}$

Step 1.6.2

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

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Step 1.6.2.1

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

$10 = A + \frac{(x - 2) (B (x - 2))}{x - 2}$

Step 1.6.2.2

Cancel the common factors.

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Step 1.6.2.2.1

Multiply by $1$ .

$10 = A + \frac{(x - 2) (B (x - 2))}{(x - 2) \cdot 1}$

Step 1.6.2.2.2

Cancel the common factor.

$10 = A + \frac{(x - 2) (B (x - 2))}{(x - 2) \cdot 1}$

Step 1.6.2.2.3

Rewrite the expression.

$10 = A + \frac{B (x - 2)}{1}$

Step 1.6.2.2.4

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

$10 = A + B (x - 2)$

Step 1.6.3

Apply the distributive property.

$10 = A + B x + B \cdot - 2$

Step 1.6.4

Move $- 2$ to the left of $B$ .

$10 = A + B x - 2 B$

Step 1.7

Reorder $A$ and $B x$ .

$10 = B x + A - 2 B$

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.

$0 = B$

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.

$10 = A - 2 B$

Step 2.3

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

$0 = B$

$10 = A - 2 B$

$0 = B$

$10 = A - 2 B$

Step 3

Solve the system of equations.

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Step 3.1

Rewrite the equation as $B = 0$ .

$B = 0$

$10 = A - 2 B$

Step 3.2

Replace all occurrences of $B$ with $0$ in each equation.

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Step 3.2.1

Replace all occurrences of $B$ in $10 = A - 2 B$ with $0$ .

$10 = A - 2 \cdot 0$

$B = 0$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $A - 2 \cdot 0$ .

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Step 3.2.2.1.1

Multiply $- 2$ by $0$ .

$10 = A + 0$

$B = 0$

Step 3.2.2.1.2

Add $A$ and $0$ .

$10 = A$

$B = 0$

$10 = A$

$B = 0$

$10 = A$

$B = 0$

$10 = A$

$B = 0$

Step 3.3

Rewrite the equation as $A = 10$ .

$A = 10$

$B = 0$

Step 3.4

Solve the system of equations.

$A = 10 B = 0$

Step 3.5

List all of the solutions.

$A = 10, B = 0$

Step 4

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

$\frac{10}{{(x - 2)}^{2}} + \frac{0}{x - 2}$