$x^{4p}$+$x^{(4q+1)}+x^{(4r+2)}+x^{(4s+ 3)} $is divisible by x^3+x^2+x+1, where ,q ,r ,s belongs to Natural numbers.
So , I did is this :
$x^3+x^2+x+1$ = $(x^2+1)(x+1)$ , So , x = +1 and -1.
Then , put in f(1) and f(-1). But I am not able to solve it further after this step.
Zeros of the polynomial $(x^2+1)(x+1)$ are $-1,\pm i$. (I hope you are acquainted with $i=\sqrt{-1}.$) Show that $f(x) = x^{4p} + x^{4q+1} + x^{4r+2} + x^{4s+3}$ satisfies $$f(-1)=f(i) = f(-i) = 0.$$ Hence conclude (e.g., by factor theorem) that the polynomial (not the number itself) $(x+1)(x^2+1)$ divides the polynomial $f(x).$
