Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given function: F(x)=5x^{5}-5x^{3}-5x^{2}+8; [-2,-1]

Answer 1

F(-2)=5(-32)-5(-8)-5(4)+8=-160+40-20+8=-132

F(-1)=-5+5-5+8=3

since one is negative and one positive, the graph must cross the x-axis somewhere between -1 and -2, by the

Intermediate Value Theorem. where it crosses the x-axis is a zero of the polynomial.

Since 3 is much closer to 0 than -132 is, odds are the zero is much closer to x=-1, maybe something like x=-1.02

IF the polynomial is a continuous function from (-2,-132) to (-1,3) then it's impossible to connect those two points without at least crossing the x-axis at least once. at that point, the y-coordinate is zero, the point(s) are (x,0) where -2<x<-1