Basic Math Examples

$\frac{z^{3} + z^{2} - 2 z - 2}{z^{2} - 4 z - 5}$

Step 1

Simplify the numerator.

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Step 1.1

Factor out the greatest common factor from each group.

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Step 1.1.1

Group the first two terms and the last two terms.

$\frac{(z^{3} + z^{2}) - 2 z - 2}{z^{2} - 4 z - 5}$

Step 1.1.2

Factor out the greatest common factor (GCF) from each group.

$\frac{z^{2} (z + 1) - 2 (z + 1)}{z^{2} - 4 z - 5}$

Step 1.2

Factor the polynomial by factoring out the greatest common factor, $z + 1$ .

$\frac{(z + 1) (z^{2} - 2)}{z^{2} - 4 z - 5}$

Step 2

Factor $z^{2} - 4 z - 5$ using the AC method.

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Step 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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 2.2

Write the factored form using these integers.

$\frac{(z + 1) (z^{2} - 2)}{(z - 5) (z + 1)}$

Step 3

Cancel the common factor of $z + 1$ .

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Step 3.1

Cancel the common factor.

$\frac{(z + 1) (z^{2} - 2)}{(z - 5) (z + 1)}$

Step 3.2

Rewrite the expression.

$\frac{z^{2} - 2}{z - 5}$