Pre-Algebra Examples

$\frac{2 s}{s^{2} - 2 s + 1} + \frac{1}{s^{2} + 6 s - 7}$

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 $1^{2}$ .

$\frac{2 s}{s^{2} - 2 s + 1^{2}} + \frac{1}{s^{2} + 6 s - 7}$

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.

$2 s = 2 \cdot s \cdot 1$

Step 1.1.3

Rewrite the polynomial.

$\frac{2 s}{s^{2} - 2 \cdot s \cdot 1 + 1^{2}} + \frac{1}{s^{2} + 6 s - 7}$

Step 1.1.4

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

$\frac{2 s}{{(s - 1)}^{2}} + \frac{1}{s^{2} + 6 s - 7}$

Step 1.2

Factor $s^{2} + 6 s - 7$ 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 $- 7$ and whose sum is $6$ .

$- 1, 7$

Step 1.2.2

Write the factored form using these integers.

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

Step 2

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

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $(s + 7) {(s - 1)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

Add $1$ and $1$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $s$ by $s$ by adding the exponents.

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Step 6.2.1

Move $s$ .

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

Step 6.2.2

Multiply $s$ by $s$ .

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

Step 6.3

Multiply $7$ by $2$ .

$\frac{2 s^{2} + 14 s + s - 1}{{(s - 1)}^{2} (s + 7)}$

Step 6.4

Add $14 s$ and $s$ .

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