High School

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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]-x^3 + 3x^2 + 5x[/tex]
D. [tex]5x^5 + 10x^4 - 25x^3[/tex]

Answer :

We are given the functions

[tex]$$
f(x) = 5x^2 \quad \text{and} \quad g(x) = x^3 + 2x^2 - 5x.
$$[/tex]

To find the product [tex]\( f(x) \cdot g(x) \)[/tex], we multiply each term of [tex]\( g(x) \)[/tex] by [tex]\( 5x^2 \)[/tex]:

1. Multiply [tex]\( 5x^2 \)[/tex] by [tex]\( x^3 \)[/tex]:

[tex]$$
5x^2 \cdot x^3 = 5x^{2+3} = 5x^5.
$$[/tex]

2. Multiply [tex]\( 5x^2 \)[/tex] by [tex]\( 2x^2 \)[/tex]:

[tex]$$
5x^2 \cdot 2x^2 = (5 \cdot 2) x^{2+2} = 10x^4.
$$[/tex]

3. Multiply [tex]\( 5x^2 \)[/tex] by [tex]\( -5x \)[/tex]:

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

Now, adding these results together we have:

[tex]$$
f(x) \cdot g(x) = 5x^5 + 10x^4 - 25x^3.
$$[/tex]

Thus, the product is

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

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