Question

Can anyone explain how the last part was gotten?

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Answer 1

do you understand what does zeros means?

Answer 2

The intermediate value theorem basically says that if a continuous function assumes values y1=f(x1) and y2=f(x2), then the function assumes any value between y1 and y2 somewhere between x1 and x2.
And a zero means when f(x)=0, x is the root (or "zero") of the function.

Answer 3

do i just look at the original equation by itself, or do i need to plug in anything before i solve?

Answer 4

Think of the x axis as a river. This river extends forever in both directions. You found that f(1) = -4 which is a negative y value. Also you found f(8) = 500 which is a positive y value.

The change from a negative y value to a positive y value means that y must be 0 at some point. This is assuming the function is continuous and there are no gaps, jumps, etc

Answer 5

the function on this interval could look like this