Basic Math Examples

$\frac{z^{2} + 10 z + 25}{z^{2} + 6 z + 5}$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rewrite $25$ as $5^{2}$ .

$\frac{z^{2} + 10 z + 5^{2}}{z^{2} + 6 z + 5}$

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.

$10 z = 2 \cdot z \cdot 5$

Step 1.3

Rewrite the polynomial.

$\frac{z^{2} + 2 \cdot z \cdot 5 + 5^{2}}{z^{2} + 6 z + 5}$

Step 1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = z$ and $b = 5$ .

$\frac{{(z + 5)}^{2}}{z^{2} + 6 z + 5}$

Step 2

Factor $z^{2} + 6 z + 5$ 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 $6$ .

$1, 5$

Step 2.2

Write the factored form using these integers.

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

Step 3

Cancel the common factor of ${(z + 5)}^{2}$ and $z + 5$ .

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Step 3.1

Factor $z + 5$ out of ${(z + 5)}^{2}$ .

$\frac{(z + 5) (z + 5)}{(z + 1) (z + 5)}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $z + 5$ out of $(z + 1) (z + 5)$ .

$\frac{(z + 5) (z + 5)}{(z + 5) (z + 1)}$

Step 3.2.2

Cancel the common factor.

$\frac{(z + 5) (z + 5)}{(z + 5) (z + 1)}$

Step 3.2.3

Rewrite the expression.

$\frac{z + 5}{z + 1}$