Pre-Algebra Examples

$v^{3} + 18 v^{2} + 81 v$

Step 1

Factor $v$ out of $v^{3} + 18 v^{2} + 81 v$ .

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Step 1.1

Factor $v$ out of $v^{3}$ .

$v \cdot v^{2} + 18 v^{2} + 81 v$

Step 1.2

Factor $v$ out of $18 v^{2}$ .

$v \cdot v^{2} + v (18 v) + 81 v$

Step 1.3

Factor $v$ out of $81 v$ .

$v \cdot v^{2} + v (18 v) + v \cdot 81$

Step 1.4

Factor $v$ out of $v \cdot v^{2} + v (18 v)$ .

$v (v^{2} + 18 v) + v \cdot 81$

Step 1.5

Factor $v$ out of $v (v^{2} + 18 v) + v \cdot 81$ .

$v (v^{2} + 18 v + 81)$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $81$ as $9^{2}$ .

$v (v^{2} + 18 v + 9^{2})$

Step 2.2

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

$18 v = 2 \cdot v \cdot 9$

Step 2.3

Rewrite the polynomial.

$v (v^{2} + 2 \cdot v \cdot 9 + 9^{2})$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = v$ and $b = 9$ .

$v {(v + 9)}^{2}$