<s>[INST] Evaluate $x+y=xy=3$ how would you evaluate $x^4+y^4$? I know how to evaluate $x^3 +y^3$ when $x+y=xy=3$, but how would you evaluate for? 

$$x^4+y^4$$

Any help would be appreciated.Thanks in advance!
 [/INST] By Vieta's formulas $x$ and $y$ are roots of the polynomial
$$ p(z)=z^2-3z+3 $$
and if $z\in\{x,y\}$ we have $z^2=3z-3$, from which $z^4=9z^2-18z+9=9z-18$ and
$$ x^4+y^4 = 9(x+y)-36 = \color{red}{-9}.$$

In general, the characteristic polynomial of the sequence $\{s_n=x^n+y^n\}_{n\geq 0}$ is exactly $p(z)$,
so $s_{n+2}=3(s_{n+1}-s_n)$. Since $s_0=2$ and $s_1=3$, it follows that $s_2=3(3-2)=3$ and
$s_3 = 3(3-3)=0$, so $s_4=3(0-3)=-9$ and $s_5=3(-9-0)=-27$, for instance.
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