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If [tex]f(x) = 0.5 x^2 - 2[/tex] and [tex]g(x) = 8 x^3 + 2[/tex], find the value of the following function:

[tex](f \cdot g)(x) = \square x^5 - \square x^3 + \square x^2 - \square[/tex]



Answer :

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]