College

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ 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.