00:01
So in this problem, we're being asked to use the intermediate value theorem to show that the given polynomial f of x has a real zero between the given integers, 1 and 5.
00:11
So essentially, what we want to show is that there's some x value in between 1 and 5, where f of that particular value, i'm going to call it c, is equal to 0.
00:22
Well, notice that f of x is a polynomial.
00:25
So that means its domain is all real numbers, meaning it's going to be a nice smooth curve.
00:29
So what the intermediate value theorem says, that if f of 1 is either positive or negative, and f of 5 is the opposite, so if f of 1 is negative, f of 5 is positive, or vice versa, that there has to be some value in between that's equal to 0.
00:46
So what we have to first do is find f of 1.
00:49
So we'll substitute in 1 for x.
00:51
So we'll have 2 times 1 to the 3rd minus 2 times 1 minus 4.
00:56
And now we'll simplify.
00:57
Well, 1 to the 3rd is 1, and 2 times 1 is 2.
01:00
So we'll have 2 for our first term...
