Precalculus Examples

$\frac{x^{2}}{x^{2} + 14 x + 49}$

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 $49$ as $7^{2}$ .

$\frac{x^{2}}{x^{2} + 14 x + 7^{2}}$

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.

$14 x = 2 \cdot x \cdot 7$

Step 1.1.3

Rewrite the polynomial.

$\frac{x^{2}}{x^{2} + 2 \cdot x \cdot 7 + 7^{2}}$

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 $b = 7$ .

$\frac{x^{2}}{{(x + 7)}^{2}}$

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 $A$ .

$\frac{A}{{(x + 7)}^{2}}$

Step 1.3

$\frac{A}{{(x + 7)}^{2}} + \frac{B}{x + 7}$

Step 1.4

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is ${(x + 7)}^{2}$ .

$\frac{x^{2} {(x + 7)}^{2}}{{(x + 7)}^{2}} = \frac{(A) {(x + 7)}^{2}}{{(x + 7)}^{2}} + \frac{(B) {(x + 7)}^{2}}{x + 7}$

Step 1.5

Cancel the common factor of ${(x + 7)}^{2}$ .

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Step 1.5.1

Cancel the common factor.

$\frac{x^{2} {(x + 7)}^{2}}{{(x + 7)}^{2}} = \frac{(A) {(x + 7)}^{2}}{{(x + 7)}^{2}} + \frac{(B) {(x + 7)}^{2}}{x + 7}$

Step 1.5.2

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

$x^{2} = \frac{(A) {(x + 7)}^{2}}{{(x + 7)}^{2}} + \frac{(B) {(x + 7)}^{2}}{x + 7}$

Step 1.6

Simplify each term.

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Step 1.6.1

Cancel the common factor of ${(x + 7)}^{2}$ .

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Step 1.6.1.1

Cancel the common factor.

$x^{2} = \frac{A {(x + 7)}^{2}}{{(x + 7)}^{2}} + \frac{(B) {(x + 7)}^{2}}{x + 7}$

Step 1.6.1.2

Divide $A$ by $1$ .

$x^{2} = A + \frac{(B) {(x + 7)}^{2}}{x + 7}$

Step 1.6.2

Cancel the common factor of ${(x + 7)}^{2}$ and $x + 7$ .

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Step 1.6.2.1

Factor $x + 7$ out of $(B) {(x + 7)}^{2}$ .

$x^{2} = A + \frac{(x + 7) (B (x + 7))}{x + 7}$

Step 1.6.2.2

Cancel the common factors.

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Step 1.6.2.2.1

Multiply by $1$ .

$x^{2} = A + \frac{(x + 7) (B (x + 7))}{(x + 7) \cdot 1}$

Step 1.6.2.2.2

Cancel the common factor.

$x^{2} = A + \frac{(x + 7) (B (x + 7))}{(x + 7) \cdot 1}$

Step 1.6.2.2.3

Rewrite the expression.

$x^{2} = A + \frac{B (x + 7)}{1}$

Step 1.6.2.2.4

Divide $B (x + 7)$ by $1$ .

$x^{2} = A + B (x + 7)$

Step 1.6.3

Apply the distributive property.

$x^{2} = A + B x + B \cdot 7$

Step 1.6.4

Move $7$ to the left of $B$ .

$x^{2} = A + B x + 7 B$

Step 1.7

Reorder $A$ and $B x$ .

$x^{2} = B x + A + 7 B$

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.

$0 = A + 7 B$

Step 2.3

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

$0 = B$

$0 = A + 7 B$

$0 = B$

$0 = A + 7 B$

Step 3

Solve the system of equations.

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Step 3.1

Rewrite the equation as $B = 0$ .

$B = 0$

$0 = A + 7 B$

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 $0 = A + 7 B$ with $0$ .

$0 = A + 7 (0)$

$B = 0$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $A + 7 (0)$ .

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Step 3.2.2.1.1

Multiply $7$ 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 $\frac{A}{{(x + 7)}^{2}} + \frac{B}{x + 7}$ with the values found for $A$ and $B$ .

$\frac{0}{{(x + 7)}^{2}} + \frac{0}{x + 7}$