Polynomials intro | Mathematics II | High School Math | Khan Academy

Let's explore the notion of a polynomial. So, this seems like a very complicated word, but if you break it down, it'll start to make sense, especially when we start to see examples of polynomials.

So, the first part of this word, let me underline it: we have "pali". This comes from Greek for "many," and you see "pali" a lot in the English language referring to the notion of many of something. So, in this case, it's many "nomials."

And "nomial" comes from Latin, from the Latin "gnomon" for "name." So, you could view this as many names, but in a mathematical context, it's really referring to many terms. We're going to talk in a little bit about what a term really is, but to get a tangible sense of what are polynomials and what are not polynomials, let me give you some examples, and then we could write some maybe more formal rules for them.

An example of a polynomial could be (10x^7 - 9x^2 + 15x^3 + 9). This is a polynomial. Another example of a polynomial is (9a^2 - 5). Even if I just have one number, even if I were to just write the number 6, that can officially be considered a polynomial. If I were to write (7x^2 - 3), let me do another variable: (7y^2 - 3y + \pi), that too would be a polynomial. So, these are examples of polynomials.

What are examples of things that are not polynomials? Well, if I were to replace the seventh power right over here with the negative seventh power, so if I were to write (10x^{-7} - 9x^2 + 15x^3 + 9), this would not be a polynomial. So, I think you might be sensing a rule here for what makes something a polynomial: that you have to have non-negative powers of your variable in each of the terms.

And I just used the word "term," so let me explain it because it'll help me explain what a polynomial is. A polynomial is something that is made up of a sum of terms.

For example, in this first polynomial, the first term is (10x^7), the second term is (-9x^2), the next term is (15x^3), and then the last term, maybe you could say the fourth term, is 9. You can see something—let me underline these—so these are all terms. This is a four-term polynomial right over here.

And you can say, "Hey, wait! This thing you wrote in red, this also has four terms," but we have to put a few more rules for it to officially be a polynomial, especially a polynomial in one variable. Each of those terms is going to be made up of a coefficient. This is the thing that multiplies the variable to some power.

So, in this first term, the coefficient is 10. And let me write this word down: "coefficient." It's another fancy word, but it's just a thing that's multiplied—in this case, times the variable, which is (x^7). So, the first coefficient is 10. The next coefficient—and actually, let me be careful here—because the second coefficient here is negative nine. So, we are looking at coefficients.

The third coefficient here is 15, and you can view this fourth term or this fourth number as the coefficient because this could be rewritten as—instead of just writing as 9, you could write it as (9x^0), and then it looks a little bit clearer, like a coefficient.

So, in general, a polynomial is the sum of a finite number of terms where each term has a coefficient, which I could represent with the letter (a), being multiplied by a variable raised to a non-negative integer power. So, this right over here is a coefficient.

It can be—if we're dealing—I don't want to get too technical. It could be a positive or negative number; it could be any real number. We have our variable, and then the exponent here has to be a non-negative integer.

So here, the reason why what I wrote in red is not a polynomial is that here I have an exponent that is a negative integer. Let's give some other examples of things that are not polynomials. So, if I were to change the second one to instead of (9a^2), if I wrote it as (9a^{1/2} - 5), this is not a polynomial because this exponent right over here is no longer an integer; it's one-half, and this is the same thing as (9\sqrt{a} - 5).

This also would not be a polynomial. Or if I were to write (9a^a - 5), also not a polynomial because here the exponent is a variable; it's not a non-negative integer. So, all of these are examples of polynomials.

So there are a few more pieces of terminology that are valuable to know. "Polynomials" is a general term for one of these expressions that has multiple terms—a finite number, so not an infinite number—and each of the terms has this form. But there's more specific terms for when you have only one term or two terms or three terms.

So, when you have one term, it's called a "monomial." So, this is a monomial. This is an example of a monomial which we could write as (6x^0). But another example of a monomial might be (10z^{15}). That's also a monomial. Your coefficient could be (\pi)—whoops, it could be (\pi)—so we could write (\pi b^5). Any of these would be monomials.

So, it's a "binomial" or "binomials" where you have two terms: "monomial" for one term, "binomial" is you have two terms. So, this right over here is a binomial. Binomial, you have two terms, and all of these are polynomials, but these are sub-classifications.

So, it's binomial; you have one, two terms. Another example of a binomial would be three (y^3 + 5y). Once again, you have two terms that have this form right over here.

Now, you'll also hear the term "trinomial." Well, trinomials are when you have three terms. "Trinomial," and this right over here is an example. This is the first term, this is the second term, and this is the third term.

Now, the next word that you will hear often in context with polynomials is the notion of the "degree" of a polynomial. You might hear people say, "What is the degree of a polynomial?" or "What is the degree of a given term of a polynomial?"

So, let's start with the degree of a given term. So, let's go to this polynomial over here. We have this first term (10x^7); the degree is the power that we're raising the variable to, so this is a seventh-degree term. The second term is a second-degree term; the third term is a third-degree term.

And you could view this constant term, which is really just 9, you could view that as—sometimes people say the constant term; sometimes people will say the zeroth-degree term. Now, if people are talking about the degree of the entire polynomial, they're going to say, "Well, what is the degree of the highest term? What is the term with the highest degree?" That degree will be the degree of the entire polynomial.

So, this first polynomial, this is a 7th-degree polynomial. This one right over here is a second-degree polynomial because it has a second-degree term, and that's the highest degree term. This right over here is a third degree; you could even say third-degree binomial because its highest degree term has degree three.

If this said (5y^7) instead of (5y), well then, it would be a 7th-degree binomial. This right over here is a 15-degree monomial; this is a second-degree trinomial.

Now, the last thing I will—or a few more things—I will introduce you to is the idea of a leading term and a leading coefficient. So, let me write this down: the notion of what it means to be leading—well, it usually means it can mean whatever is the first term, or the coefficient if you're saying "leading term" is the first term, and if you're saying "leading coefficient" is the coefficient in the first term.

But it's oftentimes associated with a polynomial being written in standard form. So, standard form is where you write the terms in degree order, starting with the highest degree term.

So, for example, what I have up here, this is not in standard form because I do have the highest degree term first, but then I should go to the next highest, which is (x^3). But here, I wrote (x^2) next, so this is not standard. If I wanted to write it in standard form, it would be (10x^7) (which is the highest degree term, that's the degree 7), then (15x^3) (which is the next highest degree), then negative (9x^2) (is the next highest degree term), and then the lowest-degree term here is (+9) or (+9x^0).

Now, this is in standard form; I've written the terms in order of decreasing degree with the highest degree first. And here, it's clear that your leading term is (10x^7) because it's the first one, and our leading coefficient here is the number 10.

So, there was a lot in that video, but hopefully, the notion of a polynomial isn't seeming too intimidating at this point. And these are really useful words to be familiar with as you continue on your math journey.