2021 Fall AMC 12B Problems/Problem 21

Problem

For real numbers $x$ , let $P(x)=1+\cos(x)+i\sin(x)-\cos(2x)-i\sin(2x)+\cos(3x)+i\sin(3x)$ where $i = \sqrt{-1}$ . For how many values of $x$ with $0\leq x<2\pi$ does $P(x)=0?$

$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$

Solution

Let $a=\cos(x)+i\sin(x)$ . Now $P(a)=1+a-a^2+a^3$ . $P(-1)=-2$ and $P(0)=1$ so there is a real number $a$ between $-1$ and $0$ . The other $a$ 's must be complex conjugates since all coefficients of the polynomial are real. The magnitude of those complex $a$ 's squared is $\frac{1}{a_1}$ which is greater than $1$ . If $x$ is real number then $a$ must have magnitude of $1$ , but all the solutions for $a$ do not have magnitude of $1$ , so the answer is $\boxed{(A) 0}

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2021 Fall AMC 12B Problems/Problem 21

Problem

Solution