<s>[INST] Polynomial : $P(a)P(a^2)P(a^3)P(a^4)$ Let $P(x) = x^3 + 2x^2+3x+4$ and $a$ be the root of equation $x^4+x^3+x^2+x+1=0$.
Find the value of $P(a)P(a^2)P(a^3)P(a^4)$
Is my answer correct ?
Since root of equation $x^4+x^3+x^2+x+1=0$ is the $5^{th}$ primitive root of 1,
so $a, a^2, a^3, a^4$ are roots of $x^4+x^3+x^2+x+1=0$ 
$P(a)P(a^4)=(a^3+2a^2+3a+4)(\frac{1}{a^3}+\frac{2}{a^2}+\frac{3}{a}+4)= 15+5a^4+5a$
Similarly, $P(a^2)P(a^3)=15+5a^3+5a^2$
$P(a)P(a^2)P(a^3)P(a^4)=(15+5a^4+5a)(15+5a^3+5a^2)=125$
 [/INST] Suppose $P(x)=x^3+2x^2+3x+4=(x-p)(x-q)(x-r)$ for some $p,q,r\in\mathbb{C}$.  Then, $$\prod_{j=1}^4\,P\left(a^j\right)=Q(p)\,Q(q)\,Q(r)\,,$$
where $Q(x):=x^4+x^3+x^2+x+1$.  Now, $$Q(x)=(x-1)\,P(x)+5\,.$$
Thus, $Q(p)=Q(q)=Q(r)=5$.  

This is actually quite a nice technique.  Let $P(x)$ and $Q(x)$ be two nonconstant polynomials in $x$ over a field $K$.  Write $\bar{P}$ and $\bar{Q}$ for the leading coefficients of $P(x)$ and $Q(x)$, respectively.  Suppose that $t_1,t_2,\ldots,t_n$ are the roots of $Q(x)$ in the algebraic closure of $K$ (with multiplicities).  Then, $$\prod_{j=1}^n\,P\left(t_j\right)=(-1)^{mn}\,\frac{\bar{P}^n}{\bar{Q}^m}\,\prod_{i=1}^m\,Q\left(s_i\right)\,,\tag{1}$$
where $s_1,s_2,\ldots,s_m$ are the roots of $P(x)$in the algebraic closure of $K$ (with multiplicities).  
If $m>n$, then write
$$P(x)=Q(x)\,A(x)+B(x)$$
for some $A(x),B(x)\in K[x]$ with $B(x)$ having degree less than $n$.  Clearly, we have $$\prod_{j=1}^n\,P\left(t_j\right)=\prod_{i=1}^m\,B\left(t_i\right)\,,\tag{2}$$
Then, we can use the paragraph below using $B(x)$ in place of $P(x)$ to simplify things even more, provided that $B(x)$ is nonconstant (or nonlinear).
If $m \leq n$, then we write $Q(x)=P(x)\,U(x)+V(x)$ for some polynomials $U(x),V(x)\in K[x]$, with $V(x)$ having degree less than $m$.  Then, from $(1)$, $$\prod_{j=1}^n\,P\left(t_j\right)=(-1)^{mn}\,\frac{\bar{P}^n}{\bar{Q}^m}\,\prod_{i=1}^m\,V\left(s_i\right)\,.\tag{3}$$
In fact, we can play this game again between $V(x)$ and $P(x)$ to make further degree reductions, provided that $V(x)$ is nonconstant (or nonlinear).

For example, let $P(x)=x^5+x^2+2x+1$ and $Q(x)=x^3-3x-2$.  Then, we see that $P(x)=A(x)\,Q(x)+B(x)$ with $A(x)=x^2+3$ and $B(x)=3x^2+11x+7$.  If $t_1$, $t_2$, and $t_3$ are the roots of $Q(x)$, then
$$\prod_{j=1}^3\,P\left(t_j\right)=\prod_{j=1}^3\,B\left(t_j\right)\,.$$
Note that $Q(x)=B(x)\,U(x)+V(x)$, where $U(x)=\frac{x}{3}-\frac{11}{9}$ and $V(x)=\frac{73}{9}x+\frac{59}{9}$.  Using $(3)$, we obtain
$$\prod_{j=1}^3\,B\left(t_j\right)=(-1)^{2\cdot 3}\,\frac{3^3}{1^2}\,\prod_{i=1}^2\,V\left(r_i\right)\,,$$
where $r_1$ and $r_2$ are the roots of $B(x)$.  That is,
$$\prod_{j=1}^3\,P\left(t_j\right)=3^3\,\prod_{i=1}^2\,V\left(r_i\right)=3^2\left(\frac{73}{9}\right)^2\left(-\frac{59}{73}-r_1\right)\left(-\frac{59}{73}-r_2\right)=3^2\left(\frac{73}{9}\right)^2\,B\left(-\frac{59}{73}\right)\,.$$
That is, $\prod_{j=1}^3\,P\left(t_j\right)=41$.  This can be easily verified as $t_1=-1$, $t_2=-1$, and $t_3=2$, so that $P\left(t_1\right)=-1$, $P\left(t_2\right)=-1$, and $P\left(t_3\right)=41$.  (I picked an easy-to-check example, of course, so this reduction method is actually more complicated than simply computing the values of $P\left(t_j\right)$'s.)
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