# How do you multiply -p^2 (p-11)?

Jul 1, 2015

Use the distributive property (also known as the distributive law or rule).

#### Explanation:

The distributive property says that for any numbers $a , b , \text{and } c$, we have:

$a \left(b + c\right) = a \cdot b + b \cdot c$

It may help to also point out that:

$p$ may be thought of as $1 p$ and
$- {p}^{2}$ is $- 1 {p}^{2}$.

Finally, $p - 11$ is the same as $p + \left(- 11\right)$

So when we distribute, we get:

$- {p}^{2} \left(p - 11\right) = - {p}^{2} \left(p + \left(- 11\right)\right)$

$= - {p}^{2} \cdot p + \left(- {p}^{2}\right) \left(- 11\right)$

$= - {p}^{3} + \left(11 {p}^{2}\right)$

$= - {p}^{3} + 11 p$

With experience, it becomes quicker to see that "when we distribute a minus sign, we have to change the signs" so, for example:
$- 2 \left(4 x - 7\right) = - 8 x + 14$
and
$- 3 \left(- 5 x - 9\right) = 15 x + 27$
and
$- x \left(- x + 4\right) = {x}^{2} - 4 x$