Question

Given the following:

f(x)=2x2−3x
𝑓
(
𝑥
)
=
2
𝑥
2
−
3
𝑥

g(x)=x2−6
𝑔
(
𝑥
)
=
𝑥
2
−
6

What is (f⋅g)(x)
(
𝑓
⋅
𝑔
)
(
𝑥
)
?

(1 point)
Responses

2x4−3x3−12x2+18x
2
𝑥
4
−
3
𝑥
3
−
12
𝑥
2
+
18
𝑥
2 x to the 4th power minus 3 x cubed minus 12 x squared plus 18 x

−13x9+18x
−
13
𝑥
9
+
18
𝑥
negative 13 x to the 9th power plus 18 x

3x2−x−6
3
𝑥
2
−
𝑥
−
6
3 x squared minus x minus 6

2x4+3x3+2x2−18

Answer 1

To find \((f \cdot g)(x)\), we need to multiply the two functions \(f(x)\) and \(g(x)\).

Given:
- \(f(x) = 2x^2 - 3x\)
- \(g(x) = x^2 - 6\)

Now, we perform the multiplication:

\[
(f \cdot g)(x) = f(x) \cdot g(x) = (2x^2 - 3x)(x^2 - 6)
\]

Using the distributive property (FOIL method), we will multiply each term in \(f(x)\) by each term in \(g(x)\):

1. \(2x^2 \cdot x^2 = 2x^4\)
2. \(2x^2 \cdot (-6) = -12x^2\)
3. \(-3x \cdot x^2 = -3x^3\)
4. \(-3x \cdot (-6) = 18x\)

Now, we combine these results:

\[
2x^4 - 3x^3 - 12x^2 + 18x
\]

This is already in the standard form. Thus, the final expression for \((f \cdot g)(x)\) is:

\[
\boxed{2x^4 - 3x^3 - 12x^2 + 18x}
\]