Proving Existence of Real Solutions for Polynomial Equations

[Janez25]

Homework Statement

Let f(x) = x^9+x^2+4. Prove: The equation f(x)=0 has at least one real solution.

Homework Equations

The Attempt at a Solution

I know that the solution lies between -2 and -1. I also know that f(-2) = -504 and f(-1) = 4. I need to know how to use the IVT to prove that there is one real solution; not sure how to do that.

 

[dx]

You've already done it. f is continuous so the intermediate value theorem applies. You showed that f(-1) = 4 and f(-2) is -504, and therefore there must be some c between -2 and -1 for which f(c) = 0 since 0 lies between 4 and -504.

 

[Janez25]

How would I write this up as a formal proof?

 

[dx]

First you point out that f is continuous, and say that the IVT applies. Then you find f(-2) and f(-1), and say that 0 lies between these two values because one is positive and one is negative. Then, by the IVT, there must be some real number c between -2 and -1 such that f(c) = 0.

 

[Janez25]

Thanks so much for all your help!