Question

The sum of the fourth powers of the roots of the equation $x^3+x+1=0$ is:

• A. $-2$
• B. $-1$
• C. $1$
• D. $2$

Answer 1

Answer: (D)

$2$

Given equation $x^3+x+1=0$
Multiply both sides by $x$, it becomes $x^4+x^2+x=0$
Since $a,b,c$ are roots of equation, then
$a^4=-a^2-a$
$b^4=-b^2-b$
$c^4=-c^2-c$
By adding above three equations, we get
$a^4+b^4+c^4=-\left[\right(a+b+c{)}^2-2(ab+bc+ca)+a+b+c]$
we get $a^4+b^4+c^4=-[-2(1\left)\right]$ $=2$ .