Use the intermediate theorem to show that the polynomial function value has a zero in the given interval

f(x)=x^5-x^4+8x^3-7x^2-17x+7; [1.6,1.8]
Find the value of f(1.6)
Find the value of f(1.8)

My calculator gave me a negative value for f(1.6)

and a positive value for f(1.8)

I wonder how the graph got from below the x-axis to above the x-axis in that little interval.

What does that mean?

mmmhh?

I plugged in x = 1.6 and got some negative value
I plugged in x = 1.8 and got some positive value
so we have two points of your graph
one is (1.6, -?) , the other is (1.8, + ?)
so one is below the x-axis , the other is above the x-axis
make a rough sketch of this situation, (it doesn't make any difference how far down or how far up you place points)
join your two points.
I see them crossing or intersecting the x-axis.
Don't we call the place where a graph cuts the x-axis, "the zero of the function" ??
I see such a point between x = 1.6 and 1.8

To use the Intermediate Value Theorem, we need to find the values of f(1.6) and f(1.8) first.

To find f(1.6), substitute x = 1.6 into the polynomial function f(x):

f(1.6) = (1.6)^5 - (1.6)^4 + 8(1.6)^3 - 7(1.6)^2 - 17(1.6) + 7

Evaluating this expression will give you the value of f(1.6).

Similarly, to find f(1.8), substitute x = 1.8 into the polynomial function f(x):

f(1.8) = (1.8)^5 - (1.8)^4 + 8(1.8)^3 - 7(1.8)^2 - 17(1.8) + 7

Evaluating this expression will give you the value of f(1.8).

Once you have the values of f(1.6) and f(1.8), you can apply the Intermediate Value Theorem to show that the polynomial function has a zero in the interval [1.6, 1.8].