00:01
So we want to show that a general nth degree polynomial has at most n minus two real roots, or real inflective points, at most n minus two involuntary points.
00:12
So here is a general polynomial.
00:14
Again, i just, you know, however, what over order it is, you have that a plus one coefficients.
00:22
Take a derivative, you just pull this down and then, you know, on and on until you get 2a2x plus a1.
00:28
A second derivative, you get n times n minus 1 times a .n times x to the n minus 2.
00:35
And then on and on until you get adhered to a 2.
00:40
And so this thing is an n minus 2 order polynomial.
00:48
And that thing, and again, i suppose there's some proof that you could prove that that thing has at most n plus n minus 1 real roots.
01:00
I'm not sure if there's a really simple way of proving that, but that's a known, a known, you know, a known thing that, you know, an nth order polynomial has at most n roots, because you could factor it as most, you know, into n x minus a or x minus b n types of terms, right? i suppose that would be one way of not really rigorous proof, but way of showing it...