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What is $(f * g)(x)$ ? $\newline$ $f(x) = x + 5$ $\newline$ $g(x) = -2x^2$ $\newline$ Write your answer as a polynomial or a rational function in simplest form. $\newline$ ______

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Power rule with rational exponents

Full solution

Q. What is

(f * g)(x)

?

\newline

f(x) = x + 5

\newline

g(x) = -2x^2

\newline

Write your answer as a polynomial or a rational function in simplest form.

\newline

______

Given Functions: We are given two functions: $\newline$ $f(x) = x + 5$ $\newline$ $g(x) = -2x^2$ $\newline$ To find the product of these two functions, denoted as $(f * g)(x)$ , we need to multiply $f(x)$ by $g(x)$ .
Multiply Functions: Let's multiply $f(x)$ by $g(x)$ :
$(f * g)(x) = f(x) * g(x)$
$= (x + 5) * (-2x^2)$
We distribute $-2x^2$ to both terms in the parentheses.
Distribute Terms: Performing the distribution: $\newline$ $(f * g)(x) = x * (-2x^2) + 5 * (-2x^2)$ $\newline$ $= -2x^3 - 10x^2$ $\newline$ We have multiplied each term correctly and combined like terms where possible.
Combine Like Terms: The final simplified form of the product of $f(x)$ and $g(x)$ is: $\newline$ $(f \cdot g)(x) = -2x^3 - 10x^2$ $\newline$ This is a polynomial in simplest form.

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