<s>[INST] How to show that a polynomial does not have real roots? For example, let's 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.
 [/INST] To show that a polynomial has no real roots, we will try to write it as an equation where the sum of some positive numbers equals a strictly negative number. As the sum of positive numbers cannot be strictly negative, there is a contradiction, which means there's no real root.
$x^8-x^7+x^2-x+15 = 0$
Subtract 15: $x^8-x^7+x^2-x = -15$
Multiply by 2: $2x^8-2x^7+2x^2-2x = -30$
Rearrange the terms: $x^8-2x^7+x^6 + x^8-2x^6+x^4 + x^6-2x^4+x^2 + x^4 + x^2-2x = -30$
Add 1: $x^8-2x^7+x^6 + x^8-2x^6+x^4 + x^6-2x^4+x^2 + x^4 + x^2-2x+1 = -29$
$(x^4-x^3)^2 + (x^4-x^2)^2 + (x^3-x)^2 + x^4 + (x-1)^2 = -29$
All terms on the left side are always positive. The sum of positive numbers cannot equal -29 so there is no real root.
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