Algebra Examples

$\frac{x^{2} + 3 x + 2}{x {(x + 2)}^{3} (x + 1)} - \frac{1}{x (x + 2) (x + 5)}$

Step 1

Simplify each term.

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Step 1.1

Factor $x^{2} + 3 x + 2$ using the AC method.

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Step 1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $3$ .

$1, 2$

Step 1.1.2

Write the factored form using these integers.

$\frac{(x + 1) (x + 2)}{x {(x + 2)}^{3} (x + 1)} - \frac{1}{x (x + 2) (x + 5)}$

Step 1.2

Cancel the common factor of $x + 1$ .

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Step 1.2.1

Cancel the common factor.

$\frac{(x + 1) (x + 2)}{x {(x + 2)}^{3} (x + 1)} - \frac{1}{x (x + 2) (x + 5)}$

Step 1.2.2

Rewrite the expression.

$\frac{x + 2}{x {(x + 2)}^{3}} - \frac{1}{x (x + 2) (x + 5)}$

Step 1.3

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

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Step 1.3.1

Multiply by $1$ .

$\frac{(x + 2) \cdot 1}{x {(x + 2)}^{3}} - \frac{1}{x (x + 2) (x + 5)}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

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

$\frac{(x + 2) \cdot 1}{(x + 2) (x {(x + 2)}^{2})} - \frac{1}{x (x + 2) (x + 5)}$

Step 1.3.2.2

Cancel the common factor.

$\frac{(x + 2) \cdot 1}{(x + 2) (x {(x + 2)}^{2})} - \frac{1}{x (x + 2) (x + 5)}$

Step 1.3.2.3

Rewrite the expression.

$\frac{1}{x {(x + 2)}^{2}} - \frac{1}{x (x + 2) (x + 5)}$

Step 2

To write $\frac{1}{x {(x + 2)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{1}{x {(x + 2)}^{2}} \cdot \frac{x + 5}{x + 5} - \frac{1}{x (x + 2) (x + 5)}$

Step 3

To write $- \frac{1}{x (x + 2) (x + 5)}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{1}{x {(x + 2)}^{2}} \cdot \frac{x + 5}{x + 5} - \frac{1}{x (x + 2) (x + 5)} \cdot \frac{x + 2}{x + 2}$

Step 4

Write each expression with a common denominator of $x (x + 5) {(x + 2)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{1}{x {(x + 2)}^{2}}$ by $\frac{x + 5}{x + 5}$ .

$\frac{x + 5}{x {(x + 2)}^{2} (x + 5)} - \frac{1}{x (x + 2) (x + 5)} \cdot \frac{x + 2}{x + 2}$

Step 4.2

Multiply $\frac{1}{x (x + 2) (x + 5)}$ by $\frac{x + 2}{x + 2}$ .

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

Step 4.3

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

$\frac{x + 5}{x {(x + 2)}^{2} (x + 5)} - \frac{x + 2}{x ({(x + 2)}^{1} (x + 2)) (x + 5)}$

Step 4.4

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

$\frac{x + 5}{x {(x + 2)}^{2} (x + 5)} - \frac{x + 2}{x ({(x + 2)}^{1} {(x + 2)}^{1}) (x + 5)}$

Step 4.5

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

$\frac{x + 5}{x {(x + 2)}^{2} (x + 5)} - \frac{x + 2}{x {(x + 2)}^{1 + 1} (x + 5)}$

Step 4.6

Add $1$ and $1$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{x + 5 - x - 1 \cdot 2}{x {(x + 2)}^{2} (x + 5)}$

Step 6.2

Multiply $- 1$ by $2$ .

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

Step 6.3

Subtract $x$ from $x$ .

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

Step 6.4

Add $0$ and $5$ .

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

Step 6.5

Subtract $2$ from $5$ .

$\frac{3}{x {(x + 2)}^{2} (x + 5)}$