00:01
All right, we're going to start with the, we're going to use the, well, let me write the polynomial px.
00:06
We have a polynomial px, x cubed, plus x squared, minus one.
00:18
And we want to prove that there's a root in between, it's a unique root, unique, it's a unique root between x equals two thirds and x equals one.
00:46
And so we're going to start with the intermediate value.
00:49
Theorem notice that p of two -thirds i'm going to type that into a calculator so i get two -thirds whoops two -thirds which i can try this again two -thirds plus two -thirds squared minus one is negative so i don't really care what the value is i care that it's just less than zero and then p of one i bet it's going to be positive i plug in one instead i get one which is graded in zero so by the ibt since p is continuous all polynomials are continuous and p goes from negative to positive and two -thirds to one, then there exists a root.
02:41
Why don't i like this? two thirds.
02:48
That's interval notation.
02:50
And so next we're going to use rolls theorem to prove that it's unique...