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1st SESSION: What are Polynomials?

You can solve the exercises proposed after you have seen the video.

TRANSCRIPTION

0:06
Hi, I’m Rob. Welcome to Math Antics.
0:09
In this video, we’re going to learn about Polynomials.
0:11
That’s a big math word for a really big concept in Algebra, so pay attention.
0:16
Now before we can understand what polynomials are, we need to learn about what mathematicians call “terms”.
0:22
In Algebra, terms are mathematical expressions that are made up of two different parts:
0:28
a number part and a variable part.
0:31
In a term, the number part and the variable part are multiplied together,
0:36
but since multiplication is implied in Algebra,
0:38
the two parts of a term are usually written right next to each other with no times symbol between them.
0:44
The number part is pretty simple… it’s just a number, like 2 or 5 or 1.4
0:49
And the number part has an official name… it’s called the “coefficient”.
0:53
Now there’s another cool math word that you can use to impress your friends at parties!
0:57
[party music, crowd noise]
0:59
…and then I said, “That’s not my wife… that’s my coefficient!”
1:04
[silence / crickets chirping]
1:07
The variable part of a term is a little more complicated.
1:11
It can be made up of one or more variables that are raised to a power.
1:14
Like… the variable part could be 'x squared'. That’s a variable raised to a power.
1:19
Or, the variable part could be just ‘y’.
1:22
If you remember what we learned in our last video, you’ll realize that that also qualifies as a variable raised to a power.
1:28
‘y’ is the same as ‘y’ to the 1st power.
1:31
But since the exponent ‘1’ doesn’t change anything, we don’t need to actually show it.
1:37
Or… the variable part of a term could be some tricky combination of variables that are raised to powers,
1:42
like ‘x squared’ times ‘y squared’.
1:44
…or ‘a’ times ‘b squared’ times ‘c cubed’.
1:48
Terms can have any number of variables like that, but the good news is that most of the time,
1:53
you’ll only need to deal with terms that have one variable. …or maybe two in complicated problems.
1:59
Oh, and there’s one thing I should point out before we move on…
2:02
if you have a term like 6y, even though it would be fine to do the multiplication the other way around and write y6,
2:09
it’s conventional to always write the number part of the term first and the variable part of the term second.
2:15
Okay, so that’s the basic idea of a term.
2:18
But there’s a little more to terms that we’ll learn in a minute.
2:20
First, let’s see how this basic idea of a term helps us understand the basic idea of a polynomial.
2:27
A polynomial is a combination of many terms.
2:30
It’s kind of like a chain of terms that are all linked together using addition or subtraction.
2:35
The terms themselves contain multiplication, but each term in a polynomial must be joined by either addition or subtraction.
2:44
And polynomials can be made from any number of terms joined together,
2:48
but there are a few specific names that are used to describe polynomials with a certain number of terms.
2:53
If there’s only one term (which isn’t really a chain) then we call it a “monomial” because the prefix “mono” means “one”.
3:01
If there are just two terms, then we call it a “binomial” because the prefix “bi” means “two”,
3:06
and if there are three terms, then we call it a “trinomial” since the prefix “tri” means “three”.
3:12
Beyond three terms, we usually just say “polynomial” since “poly” means “many”,
3:17
and in fact, it’s common to simply use the term “polynomial” even when there are just 2 or 3 terms.
3:23
Okay, so that’s the basic idea of a polynomial.
3:26
It’s a series of terms that are joined together by addition or subtraction.
3:31
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
3:41
How many terms does this polynomial have?
3:44
Well, based on what we’ve learned so far, you’re probably not quit sure.
3:48
If the terms are the parts that are joined together by addition or subtraction, then this should have three terms,
3:54
but it looks like there’s something missing with the last two terms.
3:57
This middle term is missing its number part, and this last term is missing its variable part.
4:03
That doesn’t seem to fit with our original definition of a term. What’s up with that?
4:08
Well, the middle term is easy to explain.
4:10
There really is a number part there, but it’s just ‘1’.
4:13
Do you remember how ‘1’ is always a factor of any number?
4:17
But, since multiplying by ‘1’ has no effect on a number or variable, we don’t need to show it.
4:24
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’.
4:33
Okay, but what about this last term that’s missing its variable part?
4:37
Well, that’s a little trickier. Do you remember in our last video about exponents in Algebra,
4:43
we learned that any number or variable that’s raised to the 0th power just equals ‘1’?
4:49
That means we can think of this last term as having a variable ‘x’ that’s being raised to the 0th power.
4:56
Since that would always just equal ‘1’, it’s not really a variable in the true sense of the word,
5:00
and it has no effect on the value of the term.
5:03
But it makes sense, especially if you remember the other rule from the last video.
5:07
That rule says that any number raised to the 1st power is just itself,
5:13
which helps us see that this middle term is basically the same as ‘1x’ raised to the 1st power.
5:19
Now do you see the pattern?
5:21
Each term has a number part and each term has a variable part that is raised to a power: 0, 1 and 2.
5:28
But since ‘x’ to the ‘0’ is just ‘1’,
5:31
and ‘x’ to the ‘1’ is just ‘x’,
5:33
and anything multiplied by ‘1’ is just itself,
5:37
the polynomial gets simplified so that it no longer looks exactly like the pattern it comes from.
5:43
Oh, and this last term… the one that doesn’t have a truly variable part…
5:47
it’s called a CONSTANT term because its value always stays the same.
5:52
Alright… Now that you know what a Polynomial is, let’s talk about an important property of terms and polynomials called their “degree”.
6:01
Now that might sound like the units we use to measure temperature or angles, but the degree we’re talking about here is different.
6:07
The degree of a term is determined by the power of the variable part.
6:12
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.
6:20
And in this term, the power of the variable is 3, so it’s a 3rd degree term.
6:25
Likewise, this would be a 2nd degree term and this would be a 1st degree term.
6:30
Oh, and I suppose you could call a term with no variable part a “zero degree” term,
6:35
but it’s usually just referred to as a “constant term”.
6:38
Things are a little more complicated when you have terms with more that one variable.
6:43
In that case, you add up the powers of each variable to get the degree of the term.
6:47
Since the powers in this term are 3 and 2, it’s a 5th degree term because 3 + 2 = 5.
6:54
Okay, but why do we care about the degree of terms?
6:58
Well, it’s because polynomials are often referred to by the degree of their highest term.
7:04
If a polynomial contains a 4th degree term (but no higher terms), then it’s called a “4th degree” polynomial.
7:10
But if its highest term is only a 2nd degree term, then it’s called a “2nd degree” polynomial.
7:17
Another reason that we care about the degree of the terms is that it helps us decide the arrangement of a polynomial.
7:22
We arrange the terms in a polynomial in order from the highest degree to the lowest.
7:28
…ya know, cuz, mathematicians like to keep things organized…
7:37
[mumbeling] …nice… let’s see…double check…
7:45
Perfect!
7:47
For example, this polynomial (which has 5 terms)
7:50
should be rearranged so that the highest degree term is on the left, and the lowest degree term is on the right.
7:57
But of course, not every polynomial has a term of every degree.
8:01
This is a 5th degree polynomial, but it only has 3 terms.
8:05
We should still put them in order from highest to lowest, even though it has terms that are missing.
8:11
So, the “4x to the fifth” should come first.
8:14
And then the “minus 10x”.
8:17
And finally, the “plus 8”.
8:19
By the way, it’s totally fine for a polynomial to have “missing” terms like that.
8:24
And it’s sometimes helpful to think of those missing terms as just having coefficients that are all zeros.
8:30
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.
8:38
And speaking of coefficients…
8:39
What if we need to re-arrange this polynomial so that its terms are in order from highest degree to lowest degree?
8:45
The highest degree term is ‘5x squared’ but before we just move it to the front of the polynomial,
8:52
it’s important to notice that it’s got a minus sign in front of it.
8:55
Normally when we see a minus sign, we think of subtraction, but when it comes to polynomials,
9:01
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).
9:09
In fact, instead of thinking of a polynomial as having terms that are added OR subtracted,
9:15
it’s best to think of ALL of the terms as being ADDED,
9:19
but that each term has either a POSITIVE or a NEGATIVE coefficient which is determined by the operator right in front of that term.
9:27
For example, if you have this Polynomial, you should treat it as if all of the terms are being added together,
9:33
and use the sign that’s directly in front of each term to tell you if it’s a positive or a negative term.
9:39
This first term has a coefficient of ‘negative 4’, so it’s a negative term.
9:44
The next term has a coefficient of ‘positive 6’, so it’s positive.
9:49
The next term has a coefficient of ‘negative 8’, so it’s negative.
9:54
And the constant term is just ‘positive 2’.
9:57
And recognizing positive and negative coefficients helps us a lot when
10:01
rearranging polynomials that have a mixture of positive and negative terms like our example here.
10:07
If you think of the negative sign in front of the ‘5x squared’ term as part of its coefficient,
10:13
then you’ll realize that when we move it to the front of the polynomial, the negative sign has to come with it.
10:20
It has to come with it because it’s really a NEGATIVE term.
10:23
If we don’t bring the negative sign along with it, we’ll be changing it into a positive term
10:28
which would actually change the value of the polynomial.
10:32
And in addition to helping us re-arrange them,
10:34
treating a polynomial as a combination of positive and negative terms will be very helpful when we need to simplify them,
10:42
which just so happens to be the subject of our next basic Algebra video.
10:46
Alright, we’ve learned a LOT about polynomials in this video,
10:50
and if you’re a little overwhelmed, don’t worry… it might just take some time for it all to make sense.
10:55
Remember, you can always re-watch this video a few times,
10:58
and doing some of the practice problems will help it all sink in.
11:01
As always, thanks for watching Math Antics, and I’ll see ya next time.

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