Question

select the correct answer.
which expression is equivalent to the given polynomial expression?
$(g - h)(g^2 - 3gh + 2h^2)$
\\(\circ\\) a. $g^3 - 2g^2h + gh^2 - 2h^3$
\\(\circ\\) b. $g^3 + 4g^2h^2 - 2h^3$
\\(\circ\\) c. $g^3 - 2h^3$
\\(\circ\\) d. $g^3 - 4g^2h + 5gh^2 - 2h^3$

Explanation:

Step1: Apply Distributive Property

Multiply \( g \) and \( -h \) with each term in \( (g^2 - 3gh + 2h^2) \):
\( g(g^2 - 3gh + 2h^2) - h(g^2 - 3gh + 2h^2) \)
\( = g^3 - 3g^2h + 2gh^2 - g^2h + 3gh^2 - 2h^3 \)

Step2: Combine Like Terms

Combine terms with \( g^2h \) and \( gh^2 \):

• For \( g^2h \): \( -3g^2h - g^2h = -4g^2h \) (Wait, no—wait, let's recheck. Wait, original expansion:

Wait, no, wait: \( g(g^2 - 3gh + 2h^2) = g^3 - 3g^2h + 2gh^2 \)
\( -h(g^2 - 3gh + 2h^2) = -g^2h + 3gh^2 - 2h^3 \)
Now combine like terms:
\( g^3 + (-3g^2h - g^2h) + (2gh^2 + 3gh^2) - 2h^3 \)
\( = g^3 - 4g^2h + 5gh^2 - 2h^3 \)? Wait, no, wait the options: Wait, option D is \( g^3 - 4g^2h + 5gh^2 - 2h^3 \)? Wait, no, let's re-express the original problem. Wait, the given polynomial is \( (g - h)(g^2 - 3gh + 2h^2) \). Wait, maybe I made a mistake. Wait, let's redo the distribution:

Wait, \( (g - h)(g^2 - 3gh + 2h^2) \)
= \( g(g^2 - 3gh + 2h^2) - h(g^2 - 3gh + 2h^2) \)
= \( g^3 - 3g^2h + 2gh^2 - g^2h + 3gh^2 - 2h^3 \)
Now combine \( -3g^2h - g^2h = -4g^2h \)
Combine \( 2gh^2 + 3gh^2 = 5gh^2 \)
So the result is \( g^3 - 4g^2h + 5gh^2 - 2h^3 \), which is option D. Wait, but let's check the options again. Wait, the options:

A. \( g^3 - 2g^2h + gh^2 - 2h^3 \)
B. \( g^3 + 4g^2h^2 - 2h^3 \)
C. \( g^3 - 2h^3 \)
D. \( g^3 - 4g^2h + 5gh^2 - 2h^3 \)

Yes, so after expanding and combining like terms, we get option D.

Answer:

D. \( g^3 - 4g^2h + 5gh^2 - 2h^3 \)