Trigonometry Examples

$12 p^{3} - 12 p^{2} + 3 p$

Step 1

Factor $3 p$ out of $12 p^{3} - 12 p^{2} + 3 p$ .

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Step 1.1

Factor $3 p$ out of $12 p^{3}$ .

$3 p (4 p^{2}) - 12 p^{2} + 3 p$

Step 1.2

Factor $3 p$ out of $- 12 p^{2}$ .

$3 p (4 p^{2}) + 3 p (- 4 p) + 3 p$

Step 1.3

Factor $3 p$ out of $3 p$ .

$3 p (4 p^{2}) + 3 p (- 4 p) + 3 p (1)$

Step 1.4

Factor $3 p$ out of $3 p (4 p^{2}) + 3 p (- 4 p)$ .

$3 p (4 p^{2} - 4 p) + 3 p (1)$

Step 1.5

Factor $3 p$ out of $3 p (4 p^{2} - 4 p) + 3 p (1)$ .

$3 p (4 p^{2} - 4 p + 1)$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $4 p^{2}$ as ${(2 p)}^{2}$ .

$3 p ({(2 p)}^{2} - 4 p + 1)$

Step 2.2

Rewrite $1$ as $1^{2}$ .

$3 p ({(2 p)}^{2} - 4 p + 1^{2})$

Step 2.3

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

$4 p = 2 \cdot (2 p) \cdot 1$

Step 2.4

Rewrite the polynomial.

$3 p ({(2 p)}^{2} - 2 \cdot (2 p) \cdot 1 + 1^{2})$

Step 2.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 2 p$ and $b = 1$ .

$3 p {(2 p - 1)}^{2}$