So for example, that polynomial right there is a third degree polynomial.

Why is that a third degree polynomial?

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

So that's where we get it's a third degree polynomial.

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

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

like any polynomial.

This is the standard form of the polynomial where you have it in descending order of degree.

We've simplified this polynomial.

We've simplified this polynomial.

For example, over the integers modulo , the derivative of the polynomial is the polynomial .

You can do this really for any degree polynomial dividing into any other degree polynomial.

So this one must have a third real 0, because if the third one was complex, you'd need another complex 0, and you can't have four 0's for a third degree polynomial.

The term polynomial is more general.

And then one last part to dissect the polynomial properly is to understand the coefficients of a polynomial.

This is the polynomial function below it.

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

It's given by this polynomial expression right here.

So to find the opposite of this polynomial,

This is a polynomial with only one term.

Well, we haven't done it as a polynomial.

This right here is a second degree polynomial.

The standard form of a polynomial, essentially just

I have this polynomial in the denominator here.

So since we have a polynomial here that makes this differential equation nonhomogeneous, let's guess that a particular solution is a polynomial.

If you take a second degree polynomial, take its derivatives and add and subtract, you should hopefully get another second degree polynomial.

Now we've expressed it as a simplified polynomial expression.

The first terminology is the degree of the polynomial.

So let me write a fifth degree polynomial here.

So, once again, this is a fifth degree polynomial.

The checksum is generated using a polynomial value of 0x04C11DB7.

And so I'm assuming this is a second degree polynomial.

So a binomial is just a polynomial with two terms.

We have to multiply the polynomial by negative 1 1 .

And then finally this last polynomial, we could call it.

A binomial is just a polynomial with two terms, right?

So this tells us this is a fifth degree polynomial.

So the first thing is to think about, is what would a third degree polynomial look like and what are we even talking about when we say 0's.

So we end up with this nice little fifth degree polynomial.

And I end up with a three term, second degree polynomial.

Thus if our polynomial is formula_10then the possible rational roots are all the integer divisors of a 0 ( 2) formula_11(This example is simple because the polynomial is monic (i.e.

And essentially, that's the highest exponent that we have in the polynomial.

The next thing to do, if we're going to decompose this into its components, we have to figure out the factors of the denominator right here, so that we can use those factors as the denominators in each of the components, and a third degree polynomial is much, much, much harder to factor than a second degree polynomial, normally.

Now, if we have a third degree polynomial where these three x values make it 0, we can rewrite this third degree polynomial as we can rewrite it as I'll do it in a slightly different color f of x is equal to x plus 1 you'll see why I'm doing x plus 1 instead of x minus 1 in a second x plus 1 times x minus 2 times x minus r3.

We are multiplying (10a 3) by the entire polynomial (5a 2 7a 1).