Question

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

Answer 1

Given equation, $x^4+p{x}^3+q{x}^2+rx+s=0\dots \dots \dots \dots \dots \dots \left(1\right)$

Let $a,b,c,d$ be the roots of the given equation

$(\sum a^2{)}^2=\sum a^4+2\sum a^2{b}^2$

$⟹\sum a^4=(\sum a^2{)}^2-2\sum a^2{b}^2\dots \dots \dots \dots \dots .(2)$

$(\sum a{)}^2=\sum a^2+2\sum ab$

$⟹\sum a^2=(\sum a{)}^2-2\sum ab\dots \dots \dots \dots .(3)$

$(\sum ab{)}^2=\sum a^2{b}^2+2\sum a^{2}bc+6abcd$

$⟹\sum a^2{b}^2=(\sum ab{)}^2-2\sum a^{2}bc-6abcd\dots \dots \dots .(4)$

$\sum a\sum ab=\sum a^{2}bc+3abcd$

$⟹\sum a^{2}bc=\sum a\sum ab–3abcd\dots \dots \dots .\left(5\right)$

Using (2), (3), (4) and (5), we have

$\sum a^4=\left(\right(\sum a{)}^2–2\sum ab{)}^2–2\left(\right(\sum ab{)}^2–2\sum a\sum ab)$

$=\left(\right(-p{)}^2-2q{)}^2–2\left(\right(q{)}^2–2(-p)q)$

$=p^4+4q^2-4p^{2}q-2q^2-4pq$

$=p^4+2q^2-4p^{2}q-4pq$