Let's say this is my polynomial, let me call my polynomial p of x.

I have this polynomial in the denominator here.

low order polynomial such as a plus one, when we really needed a higher order polynomial to fit the data.

Sometimes a quadratic polynomial, or just a quadratic itself, or quadratic expression, but all it means is a second degree polynomial.

Let's say I'm defining, so this is a polynomial.

So a binomial is just a polynomial with two terms.

And I don't know exactly what this third degree polynomial

I've chosen the degree d of polynomial using the test set.

And this polynomial we're going to do, we're going to keep adding terms to the polynomial, so that we can better and better approximate this function.

because you can then include all those polynomial terms of x1 and x2.

And that's what's actually called a quadratic equation, or this second degree polynomial.

Let's say you try to choose what degree polynomial to fit to data.

And they tell us a couple of the 0's of this polynomial.

Now a third degree polynomial can have as many as three 0's.

That's good because then when we run imagine that we have some kind of satisfiability algorithm that runs in polynomial time it's running on a polynomial size input, so it's not like we're running a polynomial time algorithm on an exponential size input, which would take exponential time to run.

A root of a polynomial is the x coordinate of one of its x intercepts.

After every step we're canceling out the largest degree of the polynomial we're dividing into.

look at how well my fifth order polynomial hypothesis, had done on my test set.

NP hard and therefore a polynomial time solution for that problem solves everything in NP.

Might want to review factoring a polynomial, if you found that a little confusing, that step.

The simplest polynomial is just a constant, and it would just be a horizontal line someplace.

For example, if you use polynomials this might be a high degree polynomial over here and maybe a linear function over here which is a low degree polynomial your training data error tends to go like this.

An accepting certificate is information that we can use to quickly show that the answer to the decision problem is yes, (if it actually is yes) and here, short means polynomial size and quickly means polynomial time.

In that case, given that we're fitting a high order polynomial, this is a usual overfitting setting.

And so this degree of polynomial, so the parameter is no longer fit to the test set.

like the degree polynomial to use with the learning algorithm or choose the regularization parameter for learning algorithm.

If we have a very high degree of polynomial, our training error is going to be really low.

That is, if d the degree of polynomial was too large for the data set that we have.

If you have a polynomial, you could have more than one values of x that satisfy this equation.

We can also try adding polynomial features things like x2 square x2 square and product features x1, x2.

Formerly the class NP is the set of non deterministic polynomial time decidable problems that is to say that it's a problem that can be dissolved by a program that runs in polynomial time that has non deterministic elements in it.

First, it has to be NP hard, meaning that nothing in NP is harder then X. Which is to say that if you could solve X in polynomial time, you could use it to solve everything in NP in polynomial time.

So the easiest way to approximate it is to say, well, the simplest polynomial is just a constant, right?

So here, g as usual is the sigmoid function, and we can include lots of polynomial terms like these.

All right, so now we're in a good position to define the set p of polynomial time decidable problems.

On the horizontal axis I am going to plot the degree of polynomial, so as I go the right

So in that case, I'm going to pick this fourth order polynomial model and finally what this means is that that parameter d, remember d was the degree of polynomial, right d equals 2, d equals 3, up to d equals 10.

You might want to review factoring of polynomial or completing the square if you found that step a little daunting.

We know that every problem that's in P is in NP and every problem that is NP is in EXP, that is to say any problem that you can solve in polynomial time, we can certainly solve in non deterministic polynomial time, and any problem that we can solve in a non deterministic polynomial time, we can also solve in exponential time, but here's what we don't know.

There's one other thing that we know, we do know that there really is a difference between polynomial and exponential time.

They say it's a third degree polynomial of the form ax to the third plus bx squared plus cx plus d.

And so as we increase of the greater polynomial we find typically that the training error decreases, so I'm going to write j subscript train of theta there, because our training error tends to decrease with the degree of the polynomial that we fit to the data.

So if I just wanted this one term polynomial, what would be my best approximation for this function, at least at this point?

And, if you include enough polynomial terms then, you know, maybe you can get a hypotheses that separates the positive and negative examples.

So, you should you choose a linear function, a quadratic function, a cubic function, all the way up to a 10th power polynomial?