AboutPaul Klarreich Expertise I can answer questions in basic to advanced algebra (theory of equations, complex numbers), precalculus (functions, graphs, exponential, logarithmic, and trigonometric functions and identities), basic probability, and finite mathematics, including mathematical induction.
I can also try (but not guarantee) to answer questions on Abstract Algebra
-- groups, rings, etc. and Analysis -- sequences, limits, continuity.
I won't understand specialized engineering or business jargon.
Experience I taught at a two-year college for 25 years, including all subjects from algebra to third-semester calculus.
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].
