Recently I've been studying for the AP Calculus BC test. As part of the test, you need to understand how to deal with Taylor series generally, and Maclaurin series specifically. While I think I understand most of it, there is one bit that keeps coming up which I'm having trouble understanding.
A lot of problems I see on practice tests have you find the Maclaurin polynomial of a common series (e.g., $\sin (x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} + ...$) times something else (such as $x^2$). For example, from the practice questions for the 2022 AP Live Review videos:
Write the first three nonzero terms and the general term of the Maclaurin series for $x^2 \cos(x)$.
Unfortunately, they don't solve this in class like they do with some of the other problems, but the answer sheet quotes the Maclaurin series for cos(x) and then gives the following answer:
$$ x^2 \cos(x) = x^2 - \frac{x^2 x^2}{2!} + \frac{x^2 x^2}{4!} + ... + \frac{x^2 (-1)^n x^2n}{(2n!)} $$
If I'm reading that correctly, they're multiplying each term of the Maclaurin series for $\cos{x}$ by $x^2$. Is that correct? Is there any way to solve such problems as a class (for example, say if I want to find the Maclaurin series for $f(x) = x^3 \sqrt{x} \cos{x}$)?