Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

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

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

Step 1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = s$ and $b = 1$ .

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

Step 1.2

Cancel the common factor of $s + 1$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Divide $s - 1$ by $1$ .

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

Step 2

Find the common denominator.

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Step 2.1

Write $s$ as a fraction with denominator $1$ .

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

Step 2.2

Multiply $\frac{s}{1}$ by $\frac{s - 1}{s - 1}$ .

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

Step 2.3

Multiply $\frac{s}{1}$ by $\frac{s - 1}{s - 1}$ .

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

Step 2.4

Write $- 1$ as a fraction with denominator $1$ .

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

Step 2.5

Multiply $\frac{- 1}{1}$ by $\frac{s - 1}{s - 1}$ .

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

Step 2.6

Multiply $\frac{- 1}{1}$ by $\frac{s - 1}{s - 1}$ .

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $s$ by $s$ .

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

Step 4.3

Move $- 1$ to the left of $s$ .

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

Step 4.4

Rewrite $- 1 s$ as $- s$ .

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

Step 4.5

Apply the distributive property.

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

Step 4.6

Multiply $- 1$ by $- 1$ .

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

Step 4.7

Apply the distributive property.

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

Step 4.8

Multiply $- 1$ by $1$ .

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

Step 5

Simplify by adding terms.

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Step 5.1

Combine the opposite terms in $s^{2} - s - s + 1 - s^{2} - 1$ .

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Step 5.1.1

Subtract $s^{2}$ from $s^{2}$ .

$\frac{- s - s + 1 + 0 - 1}{s - 1}$

Step 5.1.2

Add $- s - s + 1$ and $0$ .

$\frac{- s - s + 1 - 1}{s - 1}$

Step 5.1.3

Subtract $1$ from $1$ .

$\frac{- s - s + 0}{s - 1}$

Step 5.1.4

Add $- s - s$ and $0$ .

$\frac{- s - s}{s - 1}$

Step 5.2

Subtract $s$ from $- s$ .

$\frac{- 2 s}{s - 1}$

Step 5.3

Move the negative in front of the fraction.

$- \frac{2 s}{s - 1}$