Precalculus Examples

$\frac{x^{2}}{x^{2} + 10 x + 25}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

Factor using the perfect square rule.

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Step 1.1.1

Rewrite $25$ as $5^{2}$ .

$\frac{x^{2}}{x^{2} + 10 x + 5^{2}}$

Step 1.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Step 1.1.3

Rewrite the polynomial.

$\frac{x^{2}}{x^{2} + 2 \cdot x \cdot 5 + 5^{2}}$

Step 1.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 5$ .

$\frac{x^{2}}{{(x + 5)}^{2}}$

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 $A$ .

$\frac{A}{{(x + 5)}^{2}}$

Step 1.3

$\frac{A}{{(x + 5)}^{2}} + \frac{B}{x + 5}$

Step 1.4

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is ${(x + 5)}^{2}$ .

$\frac{x^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} = \frac{(A) {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(B) {(x + 5)}^{2}}{x + 5}$

Step 1.5

Cancel the common factor of ${(x + 5)}^{2}$ .

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Step 1.5.1

Cancel the common factor.

$\frac{x^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} = \frac{(A) {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(B) {(x + 5)}^{2}}{x + 5}$

Step 1.5.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{(A) {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(B) {(x + 5)}^{2}}{x + 5}$

Step 1.6

Simplify each term.

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Step 1.6.1

Cancel the common factor of ${(x + 5)}^{2}$ .

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Step 1.6.1.1

Cancel the common factor.

$x^{2} = \frac{A {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(B) {(x + 5)}^{2}}{x + 5}$

Step 1.6.1.2

Divide $A$ by $1$ .

$x^{2} = A + \frac{(B) {(x + 5)}^{2}}{x + 5}$

Step 1.6.2

Cancel the common factor of ${(x + 5)}^{2}$ and $x + 5$ .

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Step 1.6.2.1

Factor $x + 5$ out of $(B) {(x + 5)}^{2}$ .

$x^{2} = A + \frac{(x + 5) (B (x + 5))}{x + 5}$

Step 1.6.2.2

Cancel the common factors.

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Step 1.6.2.2.1

Multiply by $1$ .

$x^{2} = A + \frac{(x + 5) (B (x + 5))}{(x + 5) \cdot 1}$

Step 1.6.2.2.2

Cancel the common factor.

$x^{2} = A + \frac{(x + 5) (B (x + 5))}{(x + 5) \cdot 1}$

Step 1.6.2.2.3

Rewrite the expression.

$x^{2} = A + \frac{B (x + 5)}{1}$

Step 1.6.2.2.4

Divide $B (x + 5)$ by $1$ .

$x^{2} = A + B (x + 5)$

Step 1.6.3

Apply the distributive property.

$x^{2} = A + B x + B \cdot 5$

Step 1.6.4

Move $5$ to the left of $B$ .

$x^{2} = A + B x + 5 B$

Step 1.7

Reorder $A$ and $B x$ .

$x^{2} = B x + A + 5 B$

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.

$0 = A + 5 B$

Step 2.3

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

$0 = B$

$0 = A + 5 B$

$0 = B$

$0 = A + 5 B$

Step 3

Solve the system of equations.

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Step 3.1

Rewrite the equation as $B = 0$ .

$B = 0$

$0 = A + 5 B$

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 $0 = A + 5 B$ with $0$ .

$0 = A + 5 (0)$

$B = 0$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $A + 5 (0)$ .

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Step 3.2.2.1.1

Multiply $5$ by $0$ .

$0 = A + 0$

$B = 0$

Step 3.2.2.1.2

Add $A$ and $0$ .

$0 = A$

$B = 0$

$0 = A$

$B = 0$

$0 = A$

$B = 0$

$0 = A$

$B = 0$

Step 3.3

Rewrite the equation as $A = 0$ .

$A = 0$

$B = 0$

Step 3.4

Solve the system of equations.

$A = 0 B = 0$

Step 3.5

List all of the solutions.

$A = 0, B = 0$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{{(x + 5)}^{2}} + \frac{B}{x + 5}$ with the values found for $A$ and $B$ .

$\frac{0}{{(x + 5)}^{2}} + \frac{0}{x + 5}$