Hi, I’m Rob. Welcome to Math Antics. In this video, we’re going to learn about Polynomials.
That’s a big math word for a really big concept in Algebra, so pay attention. Now before we can understand what polynomials are, we need to learn about what mathematicians call “terms”. In Algebra, terms are mathematical expressions that are made up of two different parts: a number part and a variable part.
In a term, the number part and the variable part are multiplied together, but since multiplication is implied in Algebra, the two parts of a term are usually written right next to each other with no times symbol between them. The number part is pretty simple… it’s just a number, like 2 or 5 or 1. 4 And the number part has an official name… it’s called the “coefficient”.
Now there’s another cool math word that you can use to impress your friends at parties! [party music, crowd noise] …and then I said, “That’s not my wife… that’s my coefficient! ” [silence / crickets chirping] The variable part of a term is a little more complicated.
It can be made up of one or more variables that are raised to a power. Like… the variable part could be 'x squared'. That’s a variable raised to a power.
Or, the variable part could be just ‘y’. If you remember what we learned in our last video, you’ll realize that that also qualifies as a variable raised to a power. ‘y’ is the same as ‘y’ to the 1st power.
But since the exponent ‘1’ doesn’t change anything, we don’t need to actually show it. Or… the variable part of a term could be some tricky combination of variables that are raised to powers, like ‘x squared’ times ‘y squared’. …or ‘a’ times ‘b squared’ times ‘c cubed’.
Terms can have any number of variables like that, but the good news is that most of the time, you’ll only need to deal with terms that have one variable. …or maybe two in complicated problems. Oh, and there’s one thing I should point out before we move on… if you have a term like 6y, even though it would be fine to do the multiplication the other way around and write y6, it’s conventional to always write the number part of the term first and the variable part of the term second.
Okay, so that’s the basic idea of a term. But there’s a little more to terms that we’ll learn in a minute. First, let’s see how this basic idea of a term helps us understand the basic idea of a polynomial.
A polynomial is a combination of many terms. It’s kind of like a chain of terms that are all linked together using addition or subtraction. The terms themselves contain multiplication, but each term in a polynomial must be joined by either addition or subtraction.
And polynomials can be made from any number of terms joined together, but there are a few specific names that are used to describe polynomials with a certain number of terms. If there’s only one term (which isn’t really a chain) then we call it a “monomial” because the prefix “mono” means “one”. If there are just two terms, then we call it a “binomial” because the prefix “bi” means “two”, and if there are three terms, then we call it a “trinomial” since the prefix “tri” means “three”.
Beyond three terms, we usually just say “polynomial” since “poly” means “many”, and in fact, it’s common to simply use the term “polynomial” even when there are just 2 or 3 terms. Okay, so that’s the basic idea of a polynomial. It’s a series of terms that are joined together by addition or subtraction.
Now, let’s see a typical example of a polynomial that will help us learn a little more about terms: 3 ‘x squared’ plus ‘x’ minus 5 How many terms does this polynomial have? Well, based on what we’ve learned so far, you’re probably not quit sure. If the terms are the parts that are joined together by addition or subtraction, then this should have three terms, but it looks like there’s something missing with the last two terms.
This middle term is missing its number part, and this last term is missing its variable part. That doesn’t seem to fit with our original definition of a term. What’s up with that?
Well, the middle term is easy to explain. There really is a number part there, but it’s just ‘1’. Do you remember how ‘1’ is always a factor of any number?
But, since multiplying by ‘1’ has no effect on a number or variable, we don’t need to show it. So, if you see a term in a polynomial that has only a variable part, you know that the number part (or coefficient) of that term is just ‘1’. Okay, but what about this last term that’s missing its variable part?
Well, that’s a little trickier. Do you remember in our last video about exponents in Algebra, we learned that any number or variable that’s raised to the 0th power just equals ‘1’? That means we can think of this last term as having a variable ‘x’ that’s being raised to the 0th power.
Since that would always just equal ‘1’, it’s not really a variable in the true sense of the word, and it has no effect on the value of the term. But it makes sense, especially if you remember the other rule from the last video. That rule says that any number raised to the 1st power is just itself, which helps us see that this middle term is basically the same as ‘1x’ raised to the 1st power.
Now do you see the pattern? Each term has a number part and each term has a variable part that is raised to a power: 0, 1 and 2. But since ‘x’ to the ‘0’ is just ‘1’, and ‘x’ to the ‘1’ is just ‘x’, and anything multiplied by ‘1’ is just itself, the polynomial gets simplified so that it no longer looks exactly like the pattern it comes from.
Oh, and this last term… the one that doesn’t have a truly variable part… it’s called a CONSTANT term because its value always stays the same. Alright… Now that you know what a Polynomial is, let’s talk about an important property of terms and polynomials called their “degree”. Now that might sound like the units we use to measure temperature or angles, but the degree we’re talking about here is different.
The degree of a term is determined by the power of the variable part. For example, in this term, since the power of the variable is 4, we say that the degree of the term is 4, or that it’s a 4th degree term. And in this term, the power of the variable is 3, so it’s a 3rd degree term.
Likewise, this would be a 2nd degree term and this would be a 1st degree term. Oh, and I suppose you could call a term with no variable part a “zero degree” term, but it’s usually just referred to as a “constant term”. Things are a little more complicated when you have terms with more that one variable.
In that case, you add up the powers of each variable to get the degree of the term. Since the powers in this term are 3 and 2, it’s a 5th degree term because 3 + 2 = 5. Okay, but why do we care about the degree of terms?
Well, it’s because polynomials are often referred to by the degree of their highest term. If a polynomial contains a 4th degree term (but no higher terms), then it’s called a “4th degree” polynomial. But if its highest term is only a 2nd degree term, then it’s called a “2nd degree” polynomial.
Another reason that we care about the degree of the terms is that it helps us decide the arrangement of a polynomial. We arrange the terms in a polynomial in order from the highest degree to the lowest. …ya know, cuz, mathematicians like to keep things organized… [mumbeling] …nice… let’s see…double check… Perfect!
For example, this polynomial (which has 5 terms) should be rearranged so that the highest degree term is on the left, and the lowest degree term is on the right. But of course, not every polynomial has a term of every degree. This is a 5th degree polynomial, but it only has 3 terms.
We should still put them in order from highest to lowest, even though it has terms that are missing. So, the “4x to the fifth” should come first. And then the “minus 10x”.
And finally, the “plus 8”. By the way, it’s totally fine for a polynomial to have “missing” terms like that. And it’s sometimes helpful to think of those missing terms as just having coefficients that are all zeros.
If the coefficient of a term is zero, then the whole term has a value of zero so it wouldn’t effect the polynomial at all. And speaking of coefficients… What if we need to re-arrange this polynomial so that its terms are in order from highest degree to lowest degree? The highest degree term is ‘5x squared’ but before we just move it to the front of the polynomial, it’s important to notice that it’s got a minus sign in front of it.
Normally when we see a minus sign, we think of subtraction, but when it comes to polynomials, it’s best to think of a minus sign as a NEGATIVE SIGN that means the term right after it has a negative value (or a negative coefficient). In fact, instead of thinking of a polynomial as having terms that are added OR subtracted, it’s best to think of ALL of the terms as being ADDED, but that each term has either a POSITIVE or a NEGATIVE coefficient which is determined by the operator right in front of that term. For example, if you have this Polynomial, you should treat it as if all of the terms are being added together, and use the sign that’s directly in front of each term to tell you if it’s a positive or a negative term.
This first term has a coefficient of ‘negative 4’, so it’s a negative term. The next term has a coefficient of ‘positive 6’, so it’s positive. The next term has a coefficient of ‘negative 8’, so it’s negative.
And the constant term is just ‘positive 2’. And recognizing positive and negative coefficients helps us a lot when rearranging polynomials that have a mixture of positive and negative terms like our example here. If you think of the negative sign in front of the ‘5x squared’ term as part of its coefficient, then you’ll realize that when we move it to the front of the polynomial, the negative sign has to come with it.
It has to come with it because it’s really a NEGATIVE term. If we don’t bring the negative sign along with it, we’ll be changing it into a positive term which would actually change the value of the polynomial. And in addition to helping us re-arrange them, treating a polynomial as a combination of positive and negative terms will be very helpful when we need to simplify them, which just so happens to be the subject of our next basic Algebra video.
Alright, we’ve learned a LOT about polynomials in this video, and if you’re a little overwhelmed, don’t worry… it might just take some time for it all to make sense. Remember, you can always re-watch this video a few times, and doing some of the practice problems will help it all sink in. As always, thanks for watching Math Antics, and I’ll see ya next time.
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