Intermediate Value Theorem

[ocooper4]

A solution of the polynomial
f(x)= x^5 - 6x^4 - 12x^3 + 32x^2 + 27x - 42
lies between x = 1.7 and x = 1.8. Use the Intermediate Value Theorem to isolate the x value of this solution to within 0.01.

I have no clue where to start this.

[Guest]

Preferably using a graphing calculator or better, evaluate the function at a bunch of points between 1.7 and 1.8. You don't need exact values, just find out if it's positive or negative. Start with the endpoints f(1.7) is positive and f(1. is negative. Then selectively try intermediate points between sign changes, such as f(1.75) = -0.9 is negative, f(1.72) = +0.6 is positive, etc.

Your goal is to eventually find some [tex]r[/tex] such that [tex]f(r) \gt 0 \gt f(r+.01)[/tex]. At that point you can finish it off by saying that since polynomial functions like f are continuous, the Intermediate Value Theorem applies, and since f crosses from positive to negative over some interval, it must exactly equal 0 somewhere on that interval.