# Thread: Proving the root of a cubic polynomial

1. ## Proving the root of a cubic polynomial

How, would I go about proving the root exists for:

$x^3+5x^2-4x-1$ in the interval [0,1]

I'm a little lost as to how I should start.

Thanks.

2. ## Re: Proving the root of a cubic polynomial

Try using the Intermediate Value Theorem. Does that give you any ideas?

3. ## Re: Proving the root of a cubic polynomial

It seems like I'm not doing anything though f(0) = -1 and f(1) = 1 and all you say is therefore by the intermediate value there must exist an x such that f(x) = 0 because f(0) is negative and f(1) is positive?

Am I missing any steps in the proof? Just lean on the theorem and do nothing?

4. ## Re: Proving the root of a cubic polynomial

Originally Posted by terrorsquid
It seems like I'm not doing anything though f(0) = -1 and f(1) = 1 and all you say is therefore by the intermediate value there must exist an x such that f(x) = 0 because f(0) is negative and f(1) is positive?

Am I missing any steps in the proof? Just lean on the theorem and do nothing?
If the function goes from negative to positive, or vice versa, then it must cross the x-axis somewhere in between. That's all you need to do

5. ## Re: Proving the root of a cubic polynomial

Originally Posted by Prove It
If the function goes from negative to positive, or vice versa, then it must cross the x-axis somewhere in between. That's all you need to do
Continuous function

CB

6. ## Re: Proving the root of a cubic polynomial

Originally Posted by CaptainBlack
Continuous function

CB
That is very true, and I should have pointed it out, but since polynomials are continuous and the function given by the OP is a polynomial, I thought it went without speaking. But thanks