Expert: Paul Klarreich - 10/12/2009

Question
Hi-
I need to prove that there is an x-intercept on the interval [-1,1] for the equation P(x)=x^3+4x-1.

I tried to factor, but I got nowhere, and I tried to set it to zero and solve, and I also got nowhere. I'm so lost. Where do I start?

Anne Marie

Answer
Questioner: Anne Marie
Country: United States
Category: Advanced Math
Private: No
Subject: proving there is an x-intercept on a given interval
Question: Hi-
I need to prove that there is an x-intercept on the interval [-1,1] for the equation P(x)=x^3+4x-1.

I tried to factor, but I got nowhere, and I tried to set it to zero and solve, and I also got nowhere. I'm so lost. Where do I start?

Anne Marie
...............................................
Hi, Anne Marie,

Your problem does not say to "find the x-intercept on [-1,1]", which is what you were trying to do.  This is difficult, because it is a cubic equation and they are hard to solve.

Your problem does say to "prove that there is an x-intercept on [-1,1]".  This will be easier.

You will do the following:

Find P(-1).  Observe it is negative (-6, actually).  << You observe it out loud, of course.
Find P(1).   Observe it is positive (+4).

State the Intermediate Value Theorem.  It's on page 132 of your book. (No, no, I made that up, but it's there somewhere.)

Assert that P(x) is continuous on [-1,-1] because it is a polynomial.

Observe that  0 is between -6 and +4.

Finally apply the I.V.T. to say that P(x) must be zero somewhere in [-1,1].

If you get stuck, let me know.