Use the Intermediate Value Theorem to find at least 2 distinct intervals in which the polynomial f(x) = x4 - 2x2 + 6x - 3 has zeros.

Question

Do not find the zeros.

Answer 1

There are lots of ways to do this but perhaps one of the most straight forward ways is to simply make a table of values.

x    f(x)

-5   542
-4   197
-3   42
-2   -7
-1   -10
0    -3
1    2
2    17
3    78
4    245
5    602

The intermediate value theorem says that a continuous function (such as the polynomial in this problem) obtains all values between it's extrema in a given interval. In other words if at one point y is negative and at another y is positive than somewhere in between it had to be 0.

We look to our list and see that x=-3, f(x)=42 and at x=-2, f(x)=-7. At x=-3, f(x) is positive and at x=-2, f(x) is negative, so somewhere between x=-3 and x=-2 f(x) had to be 0, by the intermediate value theorem. Thus there is a zero somewhere in the interval (-3,-2). Can you find another interval using the same method?