Sum of cubes of the roots

#vieta#polynomials

Statement

Let $x_1$ and $x_2$ be the roots of $x^2-3x+1=0$ . Compute $x_1^3+x_2^3$ without computing the roots.

Hint

Write $x_1^3+x_2^3$ using the sum $s=x_1+x_2$ and the product $p=x_1x_2$ .

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By Vieta's formulas, $s=x_1+x_2=3$ and $p=x_1x_2=1$ .

Since $(x_1+x_2)^3=x_1^3+x_2^3+3x_1x_2(x_1+x_2)$ , we get
$x_1^3+x_2^3=s^3-3ps=27-9=18.$