College

Given the functions:

[tex]\[
\begin{array}{l}
f(x) = -5x \\
g(x) = 8x^2 - 5x - 9 \\
\end{array}
\][/tex]

Find [tex]\((f \cdot g)(x)\)[/tex].

A. [tex]\(-40x^3 - 5x - 9\)[/tex]

B. [tex]\(-40x^2 + 25x + 45x\)[/tex]

C. [tex]\(-40x^4 + 25x^3 + 45x^2\)[/tex]

D. [tex]\(-40x^3 + 25x^2 + 45x\)[/tex]

Answer :

To find [tex]\((f \cdot g)(x)\)[/tex], we need to multiply the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] together. Let's break it down:

1. Write down the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:

[tex]\[
f(x) = -5x
\][/tex]
[tex]\[
g(x) = 8x^2 - 5x - 9
\][/tex]

2. Multiply [tex]\(f(x)\)[/tex] by [tex]\(g(x)\)[/tex]:

We need to distribute [tex]\(f(x)\)[/tex] across each term in [tex]\(g(x)\)[/tex].

[tex]\[
(f \cdot g)(x) = (-5x) \cdot (8x^2 - 5x - 9)
\][/tex]

3. Distribute [tex]\(-5x\)[/tex] to each term in [tex]\(g(x)\)[/tex]:

- Multiply [tex]\(-5x\)[/tex] by [tex]\(8x^2\)[/tex]:
[tex]\[
(-5x) \cdot (8x^2) = -40x^3
\][/tex]

- Multiply [tex]\(-5x\)[/tex] by [tex]\(-5x\)[/tex]:
[tex]\[
(-5x) \cdot (-5x) = 25x^2
\][/tex]

- Multiply [tex]\(-5x\)[/tex] by [tex]\(-9\)[/tex]:
[tex]\[
(-5x) \cdot (-9) = 45x
\][/tex]

4. Combine all the results:

Combine all the terms together to get the expression for [tex]\((f \cdot g)(x)\)[/tex].

[tex]\[
(f \cdot g)(x) = -40x^3 + 25x^2 + 45x
\][/tex]

Therefore, the solution is:

[tex]\[
(f \cdot g)(x) = -40x^3 + 25x^2 + 45x
\][/tex]

This expression is the final result of multiplying [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] together.

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