How to show generally that a polynomial does not have real roots. Well, for eg lets take the polynomial $x^8-x^7+x^2-x+15$ . Here the power($n=8$) is even so it can have real roots or it might not have real roots.
Something which I thought was to find the minima and show that if the minima of $p(x)$ is greater than $0$ and $a_1$ that is the coefficient of $x^8$ are both greater than $0$ then we cannot have real roots . But in this case the derivative is $8x^7-7x^6+2x-1$ and I cannot find minima for it . So what should I do in this example . Well it is already given this polynomial does not have real roots , but I have to prove it.
Also even if I get that this does not have any real roots then is this a general method for all kinds of polynomials ?
Edit: I know Strum's theorem is one general way to solve such questions but this question is from an undergrad entrance paper and I guess a method under the reach of calculus or something similar will suffice better.