Algebra Examples

$\frac{4 b}{b^{2} - 2 b + 1} + \frac{7}{b^{2} + 1 b - 2}$

Step 1

Simplify each term.

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Step 1.1

Factor using the perfect square rule.

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Step 1.1.1

Rewrite $1$ as $1^{2}$ .

$\frac{4 b}{b^{2} - 2 b + 1^{2}} + \frac{7}{b^{2} + 1 b - 2}$

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.

$2 b = 2 \cdot b \cdot 1$

Step 1.1.3

Rewrite the polynomial.

$\frac{4 b}{b^{2} - 2 \cdot b \cdot 1 + 1^{2}} + \frac{7}{b^{2} + 1 b - 2}$

Step 1.1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = b$ and $b = 1$ .

$\frac{4 b}{{(b - 1)}^{2}} + \frac{7}{b^{2} + 1 b - 2}$

Step 1.2

Factor $b^{2} + 1 b - 2$ using the AC method.

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Step 1.2.1

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 $1$ .

$- 1, 2$

Step 1.2.2

Write the factored form using these integers.

$\frac{4 b}{{(b - 1)}^{2}} + \frac{7}{(b - 1) (b + 2)}$

Step 2

To write $\frac{4 b}{{(b - 1)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{b + 2}{b + 2}$ .

$\frac{4 b}{{(b - 1)}^{2}} \cdot \frac{b + 2}{b + 2} + \frac{7}{(b - 1) (b + 2)}$

Step 3

To write $\frac{7}{(b - 1) (b + 2)}$ as a fraction with a common denominator, multiply by $\frac{b - 1}{b - 1}$ .

$\frac{4 b}{{(b - 1)}^{2}} \cdot \frac{b + 2}{b + 2} + \frac{7}{(b - 1) (b + 2)} \cdot \frac{b - 1}{b - 1}$

Step 4

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

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Step 4.1

Multiply $\frac{4 b}{{(b - 1)}^{2}}$ by $\frac{b + 2}{b + 2}$ .

$\frac{4 b (b + 2)}{{(b - 1)}^{2} (b + 2)} + \frac{7}{(b - 1) (b + 2)} \cdot \frac{b - 1}{b - 1}$

Step 4.2

Multiply $\frac{7}{(b - 1) (b + 2)}$ by $\frac{b - 1}{b - 1}$ .

$\frac{4 b (b + 2)}{{(b - 1)}^{2} (b + 2)} + \frac{7 (b - 1)}{(b - 1) (b + 2) (b - 1)}$

Step 4.3

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

$\frac{4 b (b + 2)}{{(b - 1)}^{2} (b + 2)} + \frac{7 (b - 1)}{{(b - 1)}^{1} (b - 1) (b + 2)}$

Step 4.4

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

$\frac{4 b (b + 2)}{{(b - 1)}^{2} (b + 2)} + \frac{7 (b - 1)}{{(b - 1)}^{1} {(b - 1)}^{1} (b + 2)}$

Step 4.5

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

$\frac{4 b (b + 2)}{{(b - 1)}^{2} (b + 2)} + \frac{7 (b - 1)}{{(b - 1)}^{1 + 1} (b + 2)}$

Step 4.6

Add $1$ and $1$ .

$\frac{4 b (b + 2)}{{(b - 1)}^{2} (b + 2)} + \frac{7 (b - 1)}{{(b - 1)}^{2} (b + 2)}$

Step 5

Combine the numerators over the common denominator.

$\frac{4 b (b + 2) + 7 (b - 1)}{{(b - 1)}^{2} (b + 2)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{4 b \cdot b + 4 b \cdot 2 + 7 (b - 1)}{{(b - 1)}^{2} (b + 2)}$

Step 6.2

Multiply $b$ by $b$ by adding the exponents.

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Step 6.2.1

Move $b$ .

$\frac{4 (b \cdot b) + 4 b \cdot 2 + 7 (b - 1)}{{(b - 1)}^{2} (b + 2)}$

Step 6.2.2

Multiply $b$ by $b$ .

$\frac{4 b^{2} + 4 b \cdot 2 + 7 (b - 1)}{{(b - 1)}^{2} (b + 2)}$

Step 6.3

Multiply $2$ by $4$ .

$\frac{4 b^{2} + 8 b + 7 (b - 1)}{{(b - 1)}^{2} (b + 2)}$

Step 6.4

Apply the distributive property.

$\frac{4 b^{2} + 8 b + 7 b + 7 \cdot - 1}{{(b - 1)}^{2} (b + 2)}$

Step 6.5

Multiply $7$ by $- 1$ .

$\frac{4 b^{2} + 8 b + 7 b - 7}{{(b - 1)}^{2} (b + 2)}$

Step 6.6

Add $8 b$ and $7 b$ .

$\frac{4 b^{2} + 15 b - 7}{{(b - 1)}^{2} (b + 2)}$