Basic Math Examples

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

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

Step 1

Simplify the numerator.

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Step 1.1

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

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

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

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = z

$a = z$ and

b = 1

$b = 1$ .

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

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

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

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

Step 2

Simplify terms.

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Step 2.1

Subtract

2 z

$2 z$ from

z

$z$ .

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

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

Step 2.2

Simplify terms.

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Step 2.2.1

Cancel the common factor of

z - 1

$z - 1$ and

- z + 1

$- z + 1$ .

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Step 2.2.1.1

Factor

- 1

$- 1$ out of

z

$z$ .

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

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

Step 2.2.1.2

Rewrite

- 1

$- 1$ as

- 1 (1)

$- 1 (1)$ .

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

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

Step 2.2.1.3

Factor

- 1

$- 1$ out of

- 1 (- z) - 1 (1)

$- 1 (- z) - 1 (1)$ .

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

$1$ .

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

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

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

$1$ .

$- 1 \cdot z - 1$

$- 1 \cdot z - 1$

$- 1 \cdot z - 1$

Step 2.3

Rewrite

$- 1 z$ as

$- z$ .

$- z - 1$

$- z - 1$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$