OK, here we are, pasture 2. You all right? That wasn't too bad, the last rabbit-hole, huh?

Hmmmmm... funny, how is it there're so many tables and chairs here? And with tea things? Curiouser and curiouser.

I do declare, they look like the table where the Mad Hatter and his friends have their un-birthday parties. They were so queer, you know, they claimed it's better to have a party when it's not your birthday, because then we'd have more parties in a year. It's always tea-time for them though, and they never ever got to finishing their tea. The Mad Hatter tried to explain it to me, but all I remembered was that he tried to kill time, and time took revenge on him...

Hello, what have we here? I must have been so caught up with my story that I didn't notice them. Well, now, let me introduce you to some more people of Mathsland. Good day to you, y.

O, what a pleasant surprise! Alice! I didn't expect you to be here, and at tea-time, of all times. And who's your friend here?

My friend? Oh, the one I brought along. Well, I promised Mathsland to bring some people in, so here's one.

Wonderful! How do you do? Now, if you both would excuse me..

Actually, y, I would prefer if you show our friend here what polynomials are.

O! Why, Alice, you just know when to ask questions. My friends are here already. I think they'll show you. Come on, girls, let's entertain Alice and her friend here.

Well, to start the show, let's begin from the beginning. Myself of course. I, and all my friends here, am a monomial, a one-term expression. All my friends, by themselves, are monomials as well.

Now, that won't be interesting, will it?

No, of course, but you needn't interrupt, Alice. We're getting there. Now, let y² and I get together...

We are now a binomial. Two terms of course. And if we all get together...

That looks difficult, y. What's that?

Simple. A tri-nomial, or three terms. Look:

Three of us, right? Ha ha, fooled you. Now, let's call a special friend. Yoo hoo, handsome. We need your help... ah, wonderful, here they are.

Now if you boys would help us, and 2xy, please excuse yourself for a moment... there.

Now, polynomials are just expressions, if you recall what I've told you earlier, Alice. But if you add these two boys here, the result is an equation. What you see is a quadratic equation, because the polynomial has a degree of two. The largest power any term is raised to is two, you see. Now this

y + 4 = 0

is a linear equation. Only I am involved, while the rest take a rest and have some tea. Hmmmmm... right, now it's my turn.
 
Right, I'll take over while y has some tea. Now, these equations, we will deal with them over our course. And I hope you've realized we learnt to solve linear equations in Pasture 1. Let me introduce you to some more types of equations.
2y³ + 4y² = 0
This is a cubic equation. The highest term is y cubed.

y⁴ + 2y³ + 4y² + 8 = 0
This is a quartic or biquadratic equation. You will observe that the highest power is 4.

y⁵ + 2y³ + 4y² + 6 = 0
To complete the introduction, let me introduce you to the quintic equation, of the 5th power. However, don't worry about these too much. We won't find any cubics soon, and we may not. As for the other two, we will not deal with them. Don't fret; the last two types are very rare.

Mmmmmm... that was some tea. Well, y here. Now, don't Alice frighten you with all that. You'll find quadratics very often, and linears too. Cubics are a challenge, I can grant you that, but they may appear. Quartic and quintic equations are only for you to know they exist, ha ha.

Right, before you go, may I remind you of something: polynomials are only expressions with an integral power. That is, no roots or anything funny. In a general fashion, a polynomial is in this manner:

y, erm, I think we'd better stop here. This is getting a bit tough... Could you please give my friend something to take along?

Sure, Alice; be willing to oblige.
 

1. Polynomials are of the form:

          a0xⁿ + a1xⁿ -1 + a2xⁿ -2 +......+ an

   a0 cannot be zero, n > 0 and n is an integer.
 

2. Equations result when polynomials are equated to one another or zero. Thus:

          4x + 2 = 0; 3x² + 4x + 7 = 3x + 5

   are equations.

3. Equations are linear, quadratic, cubic, quartic or quintic, depending on the highest power any term is raised to in the equation.

5x + 6 = 0 is linear;
4x² + 5x + 6 = 0 is quadratic;
3x³ + 4x² + 5x + 6 = 0 is cubic;
2x⁴ + 3x³ + 4x² + 5x + 6 = 0 is quartic;
x⁵ + 2x⁴  + 3x³ + 4x² + 5x + 6 = 0 is quintic.