Question

1. factor $-2p^{3}+2p^{2}$.a. $p^{2}(2 - 2p)$b. $p(-2p^{2}+2)$c. $2p^{2}(-p + 1)$d. $-2p(p^{2}-p)$

Explanation:

Step1: Identify greatest common factor

The terms are $-2p^3$ and $2p^2$. The greatest common factor is $2p^2$.

Step2: Factor out the GCF

Divide each term by $2p^2$:
$\frac{-2p^3}{2p^2} = -p$, $\frac{2p^2}{2p^2} = 1$
Rewrite as $2p^2(-p + 1)$, which is equivalent to $p^2(2 - 2p)$ (multiply inside the parentheses by 2 and outside by $\frac{1}{2}$, which is a valid algebraic manipulation).

Step3: Verify other options

Option b: $p(-2p^2 + 2)$ expands to $-2p^3 + 2p$, not the original expression.
Option d: $-2p(p^2 - p)$ expands to $-2p^3 + 2p^2$, but the factored form in option a is also correct and matches the simplified factoring of the GCF when adjusted for coefficient consistency.
Option c: $2p^2(-p + 1)$ is mathematically equivalent to option a, as $2p^2(-p + 1) = p^2(2 - 2p)$.

Answer:

a. $p^2(2 - 2p)$