Question

If $a,b,c,d$ are the roots of $x^4+p{x}^3+q{x}^2+rx+s=0$, find the value of $\sum a^{2}b$.

Answer 1

Given : $a,b,c,d$ are the roots of $x^4+p{x}^3+q{x}^2+rx+s=0$

Then, we have $a+b+c+d=-p.....\left(1\right)$,
$ab+ac+ad+bc+bd+cd=q....\left(2\right)$,
$abc+abd+acd+bcd=-r....\left(3\right)$.
From these equations we have
$-pq=\sum a^{2}b+3(abc+abd+acd+bcd)$
$=\sum a^{2}b-3r$;
$\therefore \sum a^{2}b=3r-pq$.