# Proving the root of a cubic polynomial

• August 27th 2011, 11:15 AM
terrorsquid
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.
• August 27th 2011, 11:18 AM
Ackbeet
Re: Proving the root of a cubic polynomial
Try using the Intermediate Value Theorem. Does that give you any ideas?
• August 27th 2011, 08:04 PM
terrorsquid
Re: Proving the root of a cubic polynomial
It seems like I'm not doing anything though :D 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?
• August 27th 2011, 08:49 PM
Prove It
Re: Proving the root of a cubic polynomial
Quote:

Originally Posted by terrorsquid
It seems like I'm not doing anything though :D 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 :)
• August 27th 2011, 09:05 PM
CaptainBlack
Re: Proving the root of a cubic polynomial
Quote:

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
• August 28th 2011, 10:12 AM
Prove It
Re: Proving the root of a cubic polynomial
Quote:

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 :)