Finite Math Examples

$\frac{z^{2} - 16 z + 64}{z^{2} - 64}$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rewrite $64$ as $8^{2}$ .

$\frac{z^{2} - 16 z + 8^{2}}{z^{2} - 64}$

Step 1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$16 z = 2 \cdot z \cdot 8$

Step 1.3

Rewrite the polynomial.

$\frac{z^{2} - 2 \cdot z \cdot 8 + 8^{2}}{z^{2} - 64}$

Step 1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = z$ and $b = 8$ .

$\frac{{(z - 8)}^{2}}{z^{2} - 64}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $64$ as $8^{2}$ .

$\frac{{(z - 8)}^{2}}{z^{2} - 8^{2}}$

Step 2.2

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 $b = 8$ .

$\frac{{(z - 8)}^{2}}{(z + 8) (z - 8)}$

Step 3

Cancel the common factor of ${(z - 8)}^{2}$ and $z - 8$ .

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Step 3.1

Factor $z - 8$ out of ${(z - 8)}^{2}$ .

$\frac{(z - 8) (z - 8)}{(z + 8) (z - 8)}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $z - 8$ out of $(z + 8) (z - 8)$ .

$\frac{(z - 8) (z - 8)}{(z - 8) (z + 8)}$

Step 3.2.2

Cancel the common factor.

$\frac{(z - 8) (z - 8)}{(z - 8) (z + 8)}$

Step 3.2.3

Rewrite the expression.

$\frac{z - 8}{z + 8}$