Question

Let $f\left(x\right)=x^3+a{x}^2+bx+c$ and $g\left(x\right)=x^3+b{x}^2+cx+a,$ where $a,b,c$ are integers with $c\ne 0.$ Suppose that the following conditions hold:
(a) $f\left(1\right)=0;$
(b) the roots of $g\left(x\right)=0$ are the squares of the roots of $f\left(x\right)=0.$
Find the value of $a^{2013}+b^{2013}+c^{2013}$.

• A. 1
• B. 0
• C. -1
• D. 2

Answer 1

The correct option is C -1

Note that $g\left(1\right)=f\left(1\right)=0$, so 1 is a root of both f(x) and g(x).
Let p and q be the other two roots of f(x), so p${}^2$ and q${}^2$ are the other two roots of g(x).
We then get $pq=-c$ and $p^2{q}^2=-a$, so $a=-c^2$.
Also $(-a{)}^2=(p+q+1{)}^2=p^2+q^2+1+2(pq+p+q)=-b+2b=b$.
$\therefore b=c^4$.
Since $f\left(1\right)=0$ we therefore get $1+c-c^2+c^4=0$.
Factorizing, we get $(c+1)(c^3-c^2+1)=0$.
Note that $c^3-c^2+1=0$ has no integer root and hence $c=1,b=1,a=1$.
$\therefore a^{2013}+b^{2013}+c^{2013}=-1$