How do you use the Intermediate Value Theorem and synthetic division to determine whether or not the following polynomial P(x) = x^4 + 2x^3 + 2x^2 - 5x + 3 have a real zero between the numbers 0 and 1?

1 Answer
Jim H
Aug 29, 2015

You can't.

Explanation:

You can use synthetic division (and the Remainder Theorem) to find

P(0) = 3 (although using division seems like the long way to do this)

and P(1) = 3

If we one of these positive and the other negative, then, since P(x) is continuous, we could use the Intermediate Value Theorem to conclude that there is a zero in (0,1).

The Intermediate Value Theorem can never be used to conclude that there is not a zero in an interval.

We could conclude that P(x) does not have exactly one zero between 0 and 1. (It must have an even number of zeros in in (0,1).)

To help explain why, here is the graph of a different polynomial function:

graph{ x^4 + 2x^3 + 2x^2 - 5x +1 [-3.067, 3.09, -1.17, 1.91]}

Call this polynomial function Q(x)

Notice that Q(0) = 1 and also Q(1) = 1 but his function does have a zero in (0,1). (It has two of them.)

Related questions
See all questions in Intermediate Value Theorem
Impact of this question
3246 views around the world
You can reuse this answer
Creative Commons License