Algebra Examples

$\frac{1}{b + 1} + \frac{b^{2} + 2}{b^{2} - b - 2} - \frac{b}{b - 2}$

Step 1

Factor $b^{2} - b - 2$ using the AC method.

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Step 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 $- 1$ .

$- 2, 1$

Step 1.2

Write the factored form using these integers.

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $(b + 1) (b - 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{1}{b + 1}$ by $\frac{b - 2}{b - 2}$ .

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

Step 3.2

Reorder the factors of $(b - 2) (b + 1)$ .

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

Step 4

Simplify terms.

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Step 4.1

Combine the numerators over the common denominator.

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

Step 4.2

Combine the opposite terms in $\frac{b - 2 + b^{2} + 2}{(b + 1) (b - 2)} - \frac{b}{b - 2}$ .

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Step 4.2.1

Add $- 2$ and $2$ .

$\frac{b + b^{2} + 0}{(b + 1) (b - 2)} - \frac{b}{b - 2}$

Step 4.2.2

Add $b^{2}$ and $0$ .

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

Step 5

Simplify each term.

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Step 5.1

Factor $b$ out of $b + b^{2}$ .

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Step 5.1.1

Raise $b$ to the power of $1$ .

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

Step 5.1.2

Factor $b$ out of $b^{1}$ .

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

Step 5.1.3

Factor $b$ out of $b^{2}$ .

$\frac{b \cdot 1 + b \cdot b}{(b + 1) (b - 2)} - \frac{b}{b - 2}$

Step 5.1.4

Factor $b$ out of $b \cdot 1 + b \cdot b$ .

$\frac{b (1 + b)}{(b + 1) (b - 2)} - \frac{b}{b - 2}$

Step 5.2

Cancel the common factor of $1 + b$ and $b + 1$ .

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Step 5.2.1

Reorder terms.

$\frac{b (b + 1)}{(b + 1) (b - 2)} - \frac{b}{b - 2}$

Step 5.2.2

Cancel the common factor.

$\frac{b (b + 1)}{(b + 1) (b - 2)} - \frac{b}{b - 2}$

Step 5.2.3

Rewrite the expression.

$\frac{b}{b - 2} - \frac{b}{b - 2}$

Step 6

Subtract $\frac{b}{b - 2}$ from $\frac{b}{b - 2}$ .

$0$