To find the product of the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex], let's start by defining each function clearly.
Given:
[tex]\[ f(x) = 0.5 x^2 - 2 \][/tex]
[tex]\[ g(x) = 8 x^3 + 2 \][/tex]
To find [tex]\((f \cdot g)(x)\)[/tex], we need to multiply [tex]\(f(x)\)[/tex] by [tex]\(g(x)\)[/tex]:
[tex]\[(f \cdot g)(x) = (0.5 x^2 - 2) \cdot (8 x^3 + 2)\][/tex]
Now, distribute each term in [tex]\(f(x)\)[/tex] by each term in [tex]\(g(x)\)[/tex]:
1. Multiply [tex]\(0.5 x^2 \cdot 8 x^3\)[/tex]:
[tex]\[0.5 x^2 \cdot 8 x^3 = 4.0 x^5 \][/tex]
2. Multiply [tex]\(0.5 x^2 \cdot 2\)[/tex]:
[tex]\[0.5 x^2 \cdot 2 = 1.0 x^2 \][/tex]
3. Multiply [tex]\(-2 \cdot 8 x^3\)[/tex]:
[tex]\[-2 \cdot 8 x^3 = -16 x^3 \][/tex]
4. Multiply [tex]\(-2 \cdot 2\)[/tex]:
[tex]\[-2 \cdot 2 = -4 \][/tex]
Combining all these terms together, we get:
[tex]\[ (f \cdot g)(x) = 4.0 x^5 - 16 x^3 + 1.0 x^2 - 4 \][/tex]
Therefore, the correct values to fill in the blanks are:
[tex]\[
(f \cdot g)(x) = 4.0 x^5 - 16 x^3 + 1.0 x^2 - 4
\][/tex]
So your function will look like this:
[tex]\[
(f \cdot g)(x) = \boxed{4.0} x^5 - \boxed{16} x^3 + \boxed{1.0} x^2 - \boxed{4}
\][/tex]
