Question

Suppose $a, b, c$ are the roots of the cubic $x^{3}-x^{2}-2= 0$. Then the value of $a^{3}+b^{3}+c^{3}$ is _____

Answer 1

Given $a + b + c =1$
$ab + bc + ca =0$
$bcc =2$
Now $( a + b + c )^{2}=1$
$a ^{2}+ b ^{2}+ c ^{2}+2 \sum ab =1$
$\therefore a^{2}+b^{2}+c^{2}=1$
Now, $a^{3}+b^{3}+c^{3}-3 a b c=(a+b+c)\left[\sum a^{2}-\sum a b\right]$
$=1(1-0)=1$
$a^{3}+b^{3}+c^{3}=1+3 a b c=1+3 \times 2=7$