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Given [tex]f(x) = 5x^2[/tex] and [tex]g(x) = x^3 + 2x^2 - 5x[/tex], what is [tex]f(x) \cdot g(x)[/tex]?

A. [tex]x^3 + 7x^2 - 5[/tex]
B. [tex]5x^6 + 10x^4 - 25x^2[/tex]
C. [tex]5x^5 + 10x^4 - 25x^3[/tex]
D. [tex]-x^3 + 3x^2 + 5x[/tex]

Answer :

To find [tex]\( f(x) \cdot g(x) \)[/tex], we need to multiply the two given functions:

1. The function [tex]\( f(x) \)[/tex] is given as [tex]\( f(x) = 5x^2 \)[/tex].
2. The function [tex]\( g(x) \)[/tex] is given as [tex]\( g(x) = x^3 + 2x^2 - 5x \)[/tex].

We need to multiply these two expressions together:

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

Now, distribute [tex]\( 5x^2 \)[/tex] across each term in [tex]\( g(x) \)[/tex]:

- Multiply [tex]\( 5x^2 \)[/tex] by [tex]\( x^3 \)[/tex]:
[tex]\[
5x^2 \cdot x^3 = 5x^{2+3} = 5x^5
\][/tex]

- Multiply [tex]\( 5x^2 \)[/tex] by [tex]\( 2x^2 \)[/tex]:
[tex]\[
5x^2 \cdot 2x^2 = 10x^{2+2} = 10x^4
\][/tex]

- Multiply [tex]\( 5x^2 \)[/tex] by [tex]\( -5x \)[/tex]:
[tex]\[
5x^2 \cdot (-5x) = -25x^{2+1} = -25x^3
\][/tex]

Now, combine all the terms:

[tex]\[
5x^5 + 10x^4 - 25x^3
\][/tex]

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

[tex]\[ 5x^5 + 10x^4 - 25x^3 \][/tex]

Therefore, the correct answer is:

[tex]\[ 5x^5 + 10x^4 - 25x^3 \][/tex]

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