2001 SMT/Algebra Problems/Problem 10

Problem

Suppose $x$ , $y$ , $z$ satisfy $\begin{aligned} x + y + z & = 3 \\ x^{2} + y^{2} + z^{2} & = 5 \\ x^{3} + y^{3} + z^{3} & = 7 \end{aligned}$ Find $x^4+y^4+z^4$ .

Solution

Let $P_n=x^n+y^n+z^n, S_1=x+y+z, S_2=xy+yz+xz,$ and $S_3=xyz$ . From the given, we know that $P_1=S_1=3,P_2=5,$ and $P_3=7$ . Then by Newton's Formulas, we have: $P_2=P_1S_1-2S_2$ $P_3=P_2S_1-P_1S_2+3S_3$ $P_4=P_3S_1-P_2S_2+P_1S_3$ From the first, we find that $S_2=2$ . From the second, we find that $S_3=-\frac{2}{3}$ . Finally, the third yields: $P_4=7\cdot3-5\cdot2+3\cdot\left(-\frac{2}{3}\right)=21-10-2=\boxed{9}$

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2001 SMT/Algebra Problems/Problem 10

Problem

Solution