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What is $(f * g)(x)$ ? $\newline$ $f(x) = 2x$ $\newline$ $g(x) = x^2 + 2x$ $\newline$ 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) = 2x

\newline

g(x) = x^2 + 2x

\newline

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

\newline

______

Prepare for multiplication: To find the product of two functions, $f(x)$ and $g(x)$ , we multiply them together. This is denoted as $(f * g)(x) = f(x) * g(x)$ . Given $f(x) = 2x$ and $g(x) = x^2 + 2x$ , we will multiply these two expressions.
Multiply $f(x)$ by $g(x)$ : First, we write down the expressions for $f(x)$ and $g(x)$ to prepare for multiplication. $\newline$ $f(x) = 2x$ $\newline$ $g(x) = x^2 + 2x$ $\newline$ Now, we multiply $f(x)$ by $g(x)$ . $\newline$ $(2x) \times (x^2 + 2x)$
Distribute $2x$ : Distribute $2x$ across the terms in the parentheses. $2x \times x^2 + 2x \times 2x$
Perform multiplication for each term: Perform the multiplication for each term. $\newline$ $2x \times x^2 = 2x^3$ $\newline$ $2x \times 2x = 4x^2$
Combine multiplication results: Combine the results of the multiplication to get the final expression. $\newline$ $(2x \times x^2) + (2x \times 2x) = 2x^3 + 4x^2$
Final expression: The final expression $2x^3 + 4x^2$ is already in simplest form, as there are no like terms to combine and no common factors to simplify.

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