Originally Posted by

**beanus** I've spent two hours looking through my text book trying to find something that explains what this question is asking but I still have not figured out what to do. Can someone please help? Thanks!

Find three non-overlapping intervals of length 1 such that each interval contains a solution of the equation. The endpoints of your intervals should be integers.

2x^3-x^2-4x+2=0

You do not need to solve the cubic to answer this question.

From the Cauchy bound on roots we know that the absolute values of all the roots are less than or equal to 3. So make a table:

Code:

x -3 -2 -1 0 1 2 3
p(x) -49 -10 3 2 -1 6 35

Hence as the cubic changes signs between -2 and -1, between 0 and 1 and between 1 and 2 there is a root in [-2,-1], [0,1] and [1,2].

(To make this calculus, or at least pre-calculus we can quote the Intermediate Value Theorem to justify the conclusion that those intervals contain a root)

CB