Finding unknown variable in a polynomial

Question

$n^2x^{2n+3}-25nx^{n+1}+150x^7$

This polynomial has $x^2-1$ as a factor for: 1) no values of n; 2) n=10 only 3) n=15 only 4) n=10 and n=15 only

^which one's correct? Could someone please tell me how to solve this question :( so difficult

Answer 1

Just adding a little more explanation to the comments: remainder theorem approach:
If $(x^2-1)$ is a factor of the given polynomial, we have: $$n^2 x^{2n+3} -25nx^{n+1} + 150x^7 = (x^2-1)*f(x,n)$$ substituting, x=1 implies $n^2-25n+150=0$. substituting x=-1 implies $-n^2-25n(-1)^{n+1}+150(-1)^7=0$.
The solutions to the former equation are n=10 and 15. For the latter consider two cases: n=odd$\implies$ n=-10 and n=-15 are the solutions and if n=even\implies$ n=10 and n=15. n cannot be 15 due to the assumption that n is even. Hence the only common solution is n=10.

Double check your solution by substituting: For n=10:
\begin{align} n^2 x^{2n+3} -25nx^{n+1} + 150x^7 &= 50x^7(2x^{16} -5x^4+3)\\ &= 50x^7 (x^2-1)(x^2+1)*(2x^4(x^8+x^4+1)-3) \end{align}

Please do check that n=15 is not valid by substitution.