math-algebra-01

# Power Sums of a Symmetric System

Real numbers $x$, $y$, $z$ satisfy the system

$$
\begin{aligned}
x + y + z &= 6,\\
x^2 + y^2 + z^2 &= 14,\\
x^3 + y^3 + z^3 &= 36.
\end{aligned}
$$

Find the value of $x^4 + y^4 + z^4$.

Give your final answer as a single integer.

To find the value of \( x^4 + y^4 + z^4 \), we start by using the given equations and applying symmetric polynomials. We know:

\[
x + y + z = 6,
\]
\[
x^2 + y^2 + z^2 = 14,
\]
\[
x^3 + y^3 + z^3 = 36.
\]

We aim to find \( x^4 + y^4 + z^4 \). First, we use the identity for the sum of squares:

\[
x^2 + y^2 + z^2 = (x + y + z)^2 - 2(xy + yz + zx).
\]

Substituting the known values:

\[
14 = 6^2 - 2(xy + yz + zx),
\]
\[
14 = 36 - 2(xy + yz + zx),
\]
\[
2(xy + yz + zx) = 22,
\]
\[
xy + yz + zx = 11.
\]

Next, we use the identity for the sum of cubes:

\[
x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx).
\]

Substituting the known values:

\[
36 - 3xyz = 6(14 - 11),
\]
\[
36 - 3xyz = 6 \cdot 3,
\]
\[
36 - 3xyz = 18,
\]
\[
3xyz = 18,
\]
\[
xyz = 6.
\]

Now, we use Newton's sums to find \( x^4 + y^4 + z^4 \). Newton's sums relate power sums of roots of a polynomial to its coefficients. Let \( P_k = x^k + y^k + z^k \). We already have \( P_1 = 6 \), \( P_2 = 14 \), and \( P_3 = 36 \). We need to find \( P_4 \).

Newton's sums for a cubic polynomial \( t^3 + at^2 + bt + c = 0 \) give us:

\[
P_k + aP_{k-1} + bP_{k-2} + cP_{k-3} = 0.
\]

For our polynomial, we have \( a = -(x + y + z) = -6 \), \( b = xy + yz + zx = 11 \), and \( c = -xyz = -6 \). Thus, the recurrence relation becomes:

\[
P_k - 6P_{k-1} + 11P_{k-2} - 6P_{k-3} = 0.
\]

We need to find \( P_4 \). Using the known values:

\[
P_3 - 6P_2 + 11P_1 - 6P_0 = 0,
\]
where \( P_0 = 3 \) (since there are three roots).

Substituting the known values:

\[
36 - 6 \cdot 14 + 11 \cdot 6 - 6 \cdot 3 = 0,
\]
\[
36 - 84 + 66 - 18 = 0,
\]
\[
0 = 0.
\]

This confirms our values are consistent. Now we find \( P_4 \):

\[
P_4 - 6P_3 + 11P_2 - 6P_1 = 0.
\]

Substituting the known values:

\[
P_4 - 6 \cdot 36 + 11 \cdot 14 - 6 \cdot 6 = 0,
\]
\[
P_4 - 216 + 154 - 36 = 0,
\]
\[
P_4 - 98 = 0,
\]
\[
P_4 = 98.
\]

Thus, the value of \( x^4 + y^4 + z^4 \) is:

\[
\boxed{98}.
\]