2021 JMC 10 Problems/Problem 22

Problem

Let $r_1,r_2,r_3,r_4$ be the roots of $P(x)= x^4+4x^3-3x^2+2x-1.$ Suppose $Q(x)$ is the monic polynomial with all six roots in the form $r_{i}+r_{j}$ for integers $1\le i < j \le 4.$ What is the coefficient of the $x^4$ term in the polynomial $Q(x)?$

$\textbf{(A) } 32 \qquad \textbf{(B) } 36 \qquad \textbf{(C) } 42 \qquad \textbf{(D) } 48 \qquad \textbf{(E) } 56$

Solution

Note that $\sum (r_{a}+r_{b})(r_{c}+r_{d})$ has $15\cdot 4=60$ terms. We can also see that all terms are in the form $r_{i}r_{j}$ or $(r_{i})^2$ . The only way $(r_{i})^2$ are produced is $(r_{i}+r_{j})(r_{i}+r_{k})$ and there are $\tbinom{3}{2}=3$ pairs of these for a given value of $i$ .

So three copies of each $(r_{i})^2$ is produced, and there are $\tfrac{60-3\cdot 4}{6}=8$ copies of $r_{i}r_{j}$ by symmetry. By Vieta's, our desired answer is equal to $3\sum (r_{i})^2 + 8\sum r_{i}r_{j} = 42.$

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2021 JMC 10 Problems/Problem 22

Problem

Solution