Basic Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Simplify terms.

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Step 2.1

Subtract $2 z$ from $z$ .

$\frac{(z + 1) (z - 1)}{- z + 1}$

Step 2.2

Simplify terms.

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Step 2.2.1

Cancel the common factor of $z - 1$ and $- z + 1$ .

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Step 2.2.1.1

Factor $- 1$ out of $z$ .

$\frac{(z + 1) (- 1 (- z) - 1)}{- z + 1}$

Step 2.2.1.2

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{(z + 1) (- 1 (- z) - 1 (1))}{- z + 1}$

Step 2.2.1.3

Factor $- 1$ out of $- 1 (- z) - 1 (1)$ .

$\frac{(z + 1) (- 1 (- z + 1))}{- z + 1}$

Step 2.2.1.4

Cancel the common factor.

$\frac{(z + 1) (- 1 (- z + 1))}{- z + 1}$

Step 2.2.1.5

Divide $(z + 1) \cdot (- 1)$ by $1$ .

$(z + 1) \cdot (- 1)$

Step 2.2.2

Apply the distributive property.

$z \cdot - 1 + 1 \cdot - 1$

Step 2.2.3

Simplify the expression.

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Step 2.2.3.1

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

$- 1 \cdot z + 1 \cdot - 1$

Step 2.2.3.2

Multiply $- 1$ by $1$ .

$- 1 \cdot z - 1$

Step 2.3

Rewrite $- 1 z$ as $- z$ .

$- z - 1$