Vieta's Formulas (Hard)

Late AMC 10/12 through early AIME — dense computation.

Vieta's Formulas: core idea

Cubics.

Roots $r,s,t$ of $x^3-3x+1=0$ . Compute $r^2+s^2+t^2$ .

Hint: Vieta + square sum. Answer: 6.

1. If $r+s=3$ , $r^3+s^3=18$ , find $rs$ .
2. For $x^4-5x^2+4=0$ , sum of squares of all real roots.
3. $r+s=5$ , $rs=3$ . Value of $r^3+s^3$ .
4. Sum of reciprocals of roots of $2x^3-3x^2-3x+2=0$ .
5. If $\alpha+\beta=\alpha\beta=3$ , find $\alpha^2+\beta^2$ .
6. Product of all roots of $x^4+1=0$ .
7. Roots in GP for $x^3+ax^2-12x-8=0$ . Find $a$ (real middle).
8. $(x-1)(x-2)(x-3)(x-4)+1$ as quadratics — product of constant terms of monic factors?
9. Discriminant of $x^3+x-1$ (depressed form $-4p^3-27q^2$ ).