High School

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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]\(-40x^4 + 25x^3 + 45x^2\)[/tex]
B. [tex]\(-40x^2 + 25x + 45x\)[/tex]
C. [tex]\(-40x^3 - 5x - 9\)[/tex]
D. [tex]\(-40x^3 + 25x^2 + 45x\)[/tex]

Answer :

To find [tex]\((f \cdot g)(x)\)[/tex], we need to multiply the two functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] together. Here’s how you can do it step-by-step:

1. Identify the Functions:

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

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

To find [tex]\((f \cdot g)(x)\)[/tex], multiply [tex]\(f(x)\)[/tex] by [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] with each term inside the parentheses:

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

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

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

4. Combine the Results:

Combine all the products to form the polynomial:

[tex]\[
-40x^3 + 25x^2 + 45x
\][/tex]

So, the expression [tex]\((f \cdot g)(x)\)[/tex] simplifies to:

[tex]\[
-40x^3 + 25x^2 + 45x
\][/tex]

This matches one of the options provided. Therefore, the correct answer is:

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

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