2008 Mock ARML 2 Problems/Problem 5

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Al is thinking of a function, $f(x)$. He reveals to Bob that the function is a polynomial of the form $f(x) = ax^8 + bx^5 + cx^2 + dx + e$, where $a$, $b$, $c$, $d$, and $e$ are complex number coefficients. Bob wishes to determine the value of $d$. For any complex number $x$ that Bob asks about, Al will tell him the value of $f(x)$. At least how many values of $x$ must Bob ask about in order to definitively determine the value of $d$?


Note that the degree of the term with coefficient $d$, $1$, is distinct from the other degrees $\mod{3}$. We claim that $3$ values of $x$ are sufficient, namely the 3rd roots of unity.

Let $\omega = e^{2\pi/3}$. Consider $(x_1,x_2,x_3) = (1, \omega, \omega^2)$. For each of $x_1,x_2,x_3$, note that $x_i^8 = x_i^5 = x_i^2$. Thus, \begin{align*} f(1) &= (a+b+c) + d + e\\ f(\omega) &= \omega^2(a+b+c) + \omega \cdot d + e \\ f(\omega^2) &= \omega (a+b+c) + \omega^2 \cdot d + e\end{align*} This is simply a non-degenerate three-equation linear system in $a+b+c,\, d,\, e$, which will determine the value of $d$. It is not difficult to see that $1$ or $2$ values of $x$ will not suffice, so the answer is $\boxed{3}$.

See also

2008 Mock ARML 2 (Problems, Source)
Preceded by
Problem 4
Followed by
Problem 6
1 2 3 4 5 6 7 8
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