Vieta's Formulas (Medium)

AMC 10/12 mid (≈Q9–16) — multi-step.

Vieta's Formulas: core idea

Symmetric transforms.

If $r+s=5$ and $rs=3$ , find $r^3+s^3$ .

Hint: $(r+s)((r+s)^2-3rs)$ . Answer: 80.

1. If $r+s=7$ and $rs=10$ , find $r^3+s^3$ .
2. For $x^2-4x+1=0$ with roots $r,s$ , find $\dfrac1r+\dfrac1s$ .
3. If $r+s=6$ and $rs=4$ , find $r^3+s^3$ .
4. For $x^2-3x-1=0$ , find $(\alpha+\beta)^2-2\alpha\beta$ where $\alpha,\beta$ are roots.
5. If $r+s=8$ and $rs=12$ , find $r^3+s^3$ .
6. For $x^2-2x-5=0$ , find the sum of squares of the roots.
7. If $r+s=9$ and $rs=14$ , find $r^3+s^3$ .
8. For $x^2-5x+2=0$ , find $\dfrac1{\alpha}+\dfrac1{\beta}$ .
9. If $r+s=4$ and $r^2+s^2=10$ , find $rs$ .