Definitions

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Any mathematical discussion starts off with some definitions. Mathematics is a very precise subject so mathematicians try to be very precise with their terms. This can be a source of confusion to students since the results often seem wordy and convoluted. Throughout this class, we'll try to work through that by giving the mathematical definition along with an English translation and lots of examples.

In English . . .

A polynomial is an expression involving the sum of products of constants and variables raised to integer powers. An individual part of the sum is called a term. The constants by which terms are multiplied are called coefficients.

In Math . . .

Let's take that a piece at a time. A polynomial starts off as the product of some variables such as xy and x. Next we're allowed to raise the variables to integer powers like x²y and x². We can also multiply those items by numbers: 2x²y and x². Finally, we can add them together to get 2x²y + x². That gives us a basic polynomial where 2x²y and x² are the terms and 2 and 1 are the coefficients.

You might be wondering where I got the 1 from as a coefficient in my example. Remember that 1 times a number (or a variable) is just the number so when there's a 1 in front of a variable, we usually don't write it. In this case, you could "fully" write out the polynomial as 2x²y + 1x² if you wanted to make the coefficient explicit.

Here are some more examples of things that are polynomials and things that aren't.

$2x^2y$	This is a polynomial. It only has one term but that's okay.
$$\frac{x^2 + 1}{x}$$	This isn't a polynomial. You're allowed to add the variables and to multiply them but not to divide.
$$\frac{x^2 + x}{x}$$	This is a tricky one. Some people might look at this and say, "You can simplify it to x + 1 and that's a polynomial." It's true that x + 1 is the simplified version of the original but it isn't true to say that x + 1 = (x² + x) / x. To be equal, the two expressions have to have the same value for every value of x. This is true almost everywhere but not when x = 0. For that the value the left hand side is 1 but the right hand side is undefined.
3	This is a polynomial. Don't be confused by the lack of a variable. That's okay.
$x + \cos(x)$	This isn't a polynomial. Polynomials aren't allowed to have any "special" functions like ones from trigonometry, e.g. cosine, or logarithms.
$x^{1/2}$	This one looks like it has potential but it isn't. Remember that the exponents always have to be integers.

Quick Tip - Spotting a Polynomial

I think it's easiest to look at an expression and decide if it isn't a polynomial. Here are some things to look for:

It isn't a polynomial if it has variables in the denominator.
It isn't a polynomial if it has exponents that aren't positive integers.
It isn't a polynomial if it has any special functions like logarithms or trig functions.
Be careful simplifying first - especially if you eliminate a variable from the denominator.

Quick Check - Identifying Polynomials

Is the following expression a polynomial?

Video Lectures

Lectures

Polynomials

We're going to start off, as math often does, with some definitions both of polynomials and some of their key components.