The sum of the cubes of all the roots of the equation x^4-3x^3-2x^2+3x+1=0 is ____ Answer:…

Question

The sum of the cubes of all the roots of the equation $x^4-3x^3-2x^2+3x+1=0$ is \_\_\_\_

Answer: 36

Accepted Answer

The elementary symmetric functions of the roots are $e_1=3,\ e_2=-2,\ e_3=-3,\ e_4=1$ (reading off coefficients of $x^4-e_1x^3+e_2x^2-e_3x+e_4$).

By Newton's identities, with $p_k=\sum(\text{root})^k$:
$p_1=e_1=3.$
$p_2=e_1p_1-2e_2=3(3)-2(-2)=9+4=13.$
$$p_3=e_1p_2-e_2p_1+3e_3=3(13)-(-2)(3)+3(-3)=39+6-9=36.$$

Sum of cubes of roots $=p_3=36$.

Solutions · 1