For more information, please see full course syllabus of Math Analysis
For more information, please see full course syllabus of Math Analysis
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Intro to Polynomials
- A polynomial is an expression of the form
where n is a nonnegative integer, a_{n}, a_{n−1}, ..., a_{0} are all real numbers (constants), and a_{n} ≠ 0.a_{n} ·x^{n} + a_{n−1} ·x^{n−1} + …+ a_{2} ·x^{2} + a_{1} ·x + a_{0}, - If the above definition is a little hard to understand, here are the key ideas:
- It starts from some nonnegative integer n. This number is the exponent that the very first x has: x^{n}.
- It has this structure: x^{n} + x^{n−1} + …+ x^{2} + x + , where each of the blanks is filled with a number. (That's what all those a's represent.)
- The a's (blank spaces above) can potentially be 0's, causing that spot to "disappear". The only spot that's not allowed to be 0 is a_{n}: the first spot. This means our x^{n} is not allowed to disappear. (Otherwise why use n if we won't have x^{n}?)
- A polynomial function is a function that is made from a polynomial, like f(x) = x^{4} + 3x^{2} − 9x + 17.
- A polynomial equation is an equation made from a polynomial, like y = x^{4} + 3x^{2} − 9x + 17.
- While we will generally use x as the variable in polynomials, we should note that any variable can be used. Like in our work with functions, x is a commonly used variable, but there are others out there.
- The degree of a polynomial is the size of the largest exponent on a variable. If the polynomial isn't in order of largest to smallest exponents, the degree might not necessarily be the first exponent you see.
- Some types of polynomials come up often enough that they get special names. Sometimes the name is based on the degree of the polynomial:
- Cubic - Degree 3
- Quadratic - Degree 2
- Linear - Degree 1
- Constant - Degree 0
- Trinomial - 3 Terms
- Binomial - 2 Terms
- Monomial - 1 Term
- Polynomials can often be broken down into multiplicative factors by the distributive property (multiplication over parentheses). Occasionally we want to take two factors and multiply them together to expand the polynomial. In the most basic form of two binomials, we have the FOIL method:
This idea can work on larger factors or more than two as well: each term in a parenthetical group multiplies all the terms in the other parenthetical group.(a+b) (x+y) = ax + bx + ay + by. - The reverse of expanding is called factoring, which we will explore extensively in later lessons.
- The long-term behavior of a polynomial is determined by the term with the largest exponent and whether or not that term has a positive or negative coefficient. This is called the Leading Coefficient Test. [To visually see what happens, check out the video for the various images. In general, though, you can imagine what ±x^{2} and ±x^{3} would do.]
Intro to Polynomials
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
- Intro
- Introduction
- Definition of a Polynomial
- Starting Integer
- Structure of a Polynomial
- The a Constants
- Polynomial Function
- Polynomial Equation
- Polynomials with Different Variables
- Degree
- Special Names for Polynomials
- Distributive Property (aka 'FOIL')
- Long-Term Behavior of Polynomials
- Examples
- Controlling Term--Term with the Largest Exponent
- Positive and Negative Coefficients on the Controlling Term
- Leading Coefficient Test
- Even Degree, Positive Coefficient
- Even Degree, Negative Coefficient
- Odd Degree, Positive Coefficient
- Odd Degree, Negative Coefficient
- Example 1
- Example 2
- Example 3
- Example 4
- Intro 0:00
- Introduction 0:04
- Definition of a Polynomial 1:04
- Starting Integer
- Structure of a Polynomial
- The a Constants
- Polynomial Function
- Polynomial Equation
- Polynomials with Different Variables
- Degree 6:23
- Informal Definition
- Find the Largest Exponent Variable
- Quick Examples
- Special Names for Polynomials 8:59
- Based on the Degree
- Based on the Number of Terms
- Distributive Property (aka 'FOIL') 11:37
- Basic Distributive Property
- Distributing Two Binomials
- Longer Parentheses
- Reverse: Factoring
- Long-Term Behavior of Polynomials 17:48
- Examples
- Controlling Term--Term with the Largest Exponent
- Positive and Negative Coefficients on the Controlling Term
- Leading Coefficient Test 22:07
- Even Degree, Positive Coefficient
- Even Degree, Negative Coefficient
- Odd Degree, Positive Coefficient
- Odd Degree, Negative Coefficient
- Example 1 25:11
- Example 2 27:16
- Example 3 31:16
- Example 4 34:41
Math Analysis Online
Transcription: Intro to Polynomials
Hi--welcome back to Educator.com.0000
Today, we are going to have an introduction to polynomials.0002
By this point, you have seen polynomials, even if you don't remember the name, countless times in previous courses.0005
As a brief reminder, they are the ones that look like x^{2} - 2x + 9,0010
or maybe 3x^{5} - 8x^{3} + 10x^{2} + x + 47.0014
This stuff looks familiar; now, you might wonder why you have spent so much time on them before,0019
and why we are studying them yet again in another course.0024
In short, it is because polynomials are ridiculously, absurdly useful.0028
They come up in every branch of science, from physics to medicine to economics.0033
They are going to be important if you are going to do engineering work; they are going to be important if you are going to do computer programming work.0038
They are going to be important for pretty much anything you want to do.0043
If you want to study higher-level mathematics, they are going to be important in that, too.0045
Polynomials are very important; they are going to be important in any branch of science, and in anything that is in higher, deeper levels of mathematics.0048
So, that is why they keep drilling them for all these years--because you really have to understand polynomials0056
for a huge number of things, so it is really important to get a good grasp on it now.0061
A polynomial: what is a polynomial? Formally, we define it as an expression of the form0066
a_{n}x^{n} + a_{n - 1}x^{n - 1} +...+ a_{2} times x^{2} + a_{1}x + a_{0}.0071
And now, don't worry; these little things down here we just call the subscripts, which just means to say that there is a,0082
but then there are many different a's; there is a_{n}, a_{n - 1}, a_{n - 2}, and so on and so on...0089
a_{2}, a_{1}, a_{0}...just many different a's.0096
What this expression means: we have that n is a non-negative integer, and all of our a's0100
(the a_{n}, a_{n - 1}, and so on, up until a_{0}), are all real numbers,0106
which is to say that they are just constants; and finally, a_{n} itself, the first one,0110
the one at the very front, is not equal to 0.0115
Now, that might seem a little complex in its formal definition; but don't worry;0119
we are about to explain what is going on, so we can really understand what a polynomial is.0122
So, our expression, once again, was a_{n}x^{n} + a_{n - 1}x^{n - 1},0126
and so on and so on...a_{1}x + a_{0}.0131
The first thing that we want to get to is: we want to start with this non-negative integer, n.0135
This n is really important; that n can be any number...something like 1 or 5 or 968.0140
It is just the exponent that the very first x has; so we could have x^{1}0147
(which we would normally write as just x), and then other stuff after it.0152
Or we could have x^{5}, and then other stuff after it; or we could have x^{968}, and then other stuff after it.0155
The n is basically our starting point--what is our starting exponent going to be?0165
Then, we have this structure: _x^{n} + _x^{n - 1} + _x^{n - 2}...so on and so on,0170
until finally we get _x^{2} + _x + _.0179
If we took x^{5}, then we would have _x^{5} + _x^{4} + _x^{3} + _x^{2} + _x + _.0184
We just fill in those blanks with numbers.0197
That is what all of these a's represent; these are our blanks, down below.0200
They are the things that we are filling in; the a's represent those blanks.0207
They are just a number that is going to get stuffed into that place.0210
And finally, the a's (blank spaces) above can be, potentially, zeroes; so if we had a 0 here, we would just knock out the whole thing.0215
And we would pretend it wasn't there; we would read it as x^{n} +...and then x^{n - 2} would be next.0223
If we have _x^{2} + _x + _, and we have 5x^{2} + 0x + 3,0228
we would probably just read this as 5x^{2} + 3.0238
So, if we have an a as a 0, it can cause that spot to just disappear.0241
Now, the only spot that is not allowed to disappear is a_{n}: a_{n}, the first spot, this one up here, is not allowed to be 0.0247
Why not? Because, if it was 0, then our x^{n} would just disappear.0259
If we were able to have 0, then it would be gone; and so, if it is gone, our x^{n} would disappear,0263
at which point, why did we choose n in the first place, if we are not even going to have x^{n} show up?0269
So, since we want to use n (that is why we chose n), we can't have our very first spot disappear and get rid of that n.0274
And that is it--that is a polynomial: _x to the exponent, plus _x to the other exponent, plus blank...and so on and so forth.0280
That is pretty much just the structure of a polynomial.0287
If you can remember that, that is the important part.0289
While a polynomial is technically just an expression, like, for example, x^{4} + 3x^{2} - 9x + 17--0292
a polynomial is just this expression of _x to the exponent + _x to the exponent + _x to the exponent--0299
that is all it is--just that structure of _x to the exponent--we normally use them to make functions or equations.0306
So, a polynomial function is just a function that has been made out of a polynomial.0314
A polynomial function is a function that is equal to some polynomial.0317
And a polynomial equation is just an equation made out of it, as well.0322
So, we could have y = polynomial, or we could have function = polynomial.0328
That is it; also, while we will generally use x as the variable in polynomials, we should note that any variable can be used.0333
Any variable can be used; the important thing is that we are just following this _something to the exponent structure.0341
Like in our work with functions, we normally use f(x); but there is no reason that we have to use x.0346
x is a commonly-used variable, but it is not the only one out there.0352
There are others out there; so all of the below are just as valid as x^{4} + 3x^{2} - 9x + 17.0356
We could have z^{4} + 3z^{2} - 9z + 17--0363
representing the same thing; but instead, now we have a different variable being the placeholder.0368
Or we could have l to the fourth and more things, or θ to the fourth.0372
Any symbol can be our placeholder; we just want something that is being that placeholder, and being raised to an exponent.0376
The degree of the polynomial is the value of n in this expression; it is whatever our highest exponent is at the front.0385
Informally, we just want to see it as...the degree of the largest exponent on a variable.0391
So, that is what we want to think of degree as: the largest exponent on one of our variables.0399
If the polynomial isn't in order of largest to smallest exponents...0404
Normally we are in order--we go to n, and then next we are at n - 1, and then next we would be at x^{n - 2}, and so on0407
and so on and so on, until eventually we got to x^{2}, and then x^{1}, and then...0415
although you might not remember this from exponent work before, x^{0}, which we will talk about0423
in exponents later on--but the point is that we keep lowering the exponent--0428
we keep going and going and going, until we are finally at a constant.0431
But if our polynomial isn't in order of largest to smallest exponents, the degree might not necessarily be the very first one that you see.0435
It might not necessarily be the one at the very beginning; it could be somewhere in the middle,0442
if we aren't necessarily in that order of largest to smallest exponents.0446
The important thing is just to find the largest exponent on a variable; and that is your degree.0450
Let's see some examples: we could have a polynomial x^{2} + 2x + 1.0455
We look at this one, and we say, "Oh, the largest exponent on anything is that 2"; so we get a degree of 2.0460
We look at this one, 5x + 3; and the biggest one here is just this x.0468
What is its exponent? The exponent of anything is just to the 1, if it doesn't have something already, so we get 1.0471
We look at the next one: 7x^{3} - 4x^{47} + 8.0479
The one at the front is x^{3}, but it isn't going to be our degree.0483
The degree ends up being...this one isn't in our usual order; it isn't in that general form0487
of x^{n} and then x^{n - 1} and then x^{n - 2}; this one is out of order.0492
But that doesn't mean that we can't find its degree; we just look through.0497
We look at all of our x's, and we end up seeing that 47 is the largest exponent on any of our variables.0501
And so, it is 47 that is our degree.0507
Finally, the last one might be a little bit confusing, as well.0509
We see this one, and we think, "Oh, x^{3}...wait, there is an even larger exponent here."0512
We have 3^{5}, but 3 is not an x; it is not a variable.0516
So, since it is not a variable, it is out of the running, which leaves us with x^{3} as what we have.0524
And so, the degree of that is 3; so you are looking for a variable (make sure it is a variable) with the highest, largest exponent.0530
And that is your degree for a polynomial.0537
Since a lot of different polynomials come up very often, we have some special names for them.0541
Some types of polynomials get special names, and so we want to know them.0545
They are not super important to remember, although quadratic will come up so often, it is definitely going to be burned into your memory.0548
It is not super important to absolutely remember these; but they will come up.0555
And so, you want to know them, because you might have to know these vocabulary words.0558
You can figure out what name to use, based on the degree of a polynomial, for these ones.0562
A cubic is a degree 3 polynomial; this one has a degree of 3 here; or 5x^{3} - 3x^{2} + 27, once again, has a degree of 3.0566
A quadratic has a degree of 2; so it is x^{2} + x + 1 or -17x^{2} + 20x - √2.0577
A linear has a degree of 1: x^{1}, πx^{1}...0585
And then finally, a constant is just a degree 0 polynomial, which is to say it has no variables in it at all.0592
So, 1 has no variables; 5,111,723 still has no variable--there is no x here, so since there is no x, we have degree 0.0599
We can also talk about a polynomial based on the number of terms that make it up.0613
Once again, it is not super important to have this really memorized; but you want to be familiar with0616
and aware of these vocabulary terms, because they will show up now and then.0620
A trinomial is something that has three terms; we can remember this from trinomial,0624
like a tricycle or a triangle--they are all things having to do with the number 3.0628
x^{2} + x + 1: the squared isn't so much the important part as the x^{2}; we have three things.0633
47x^{9} + x^{3} + 2: the degree no longer matters.0642
It is not about the degree, so I really should not have accidentally circled that 2...x^{2}...0648
It is just the number of things we have: 47x^{9} + x^{3} + 2...0657
A binomial is something that has two terms; x and then 1, or -52x^{7} and 892x.0661
It doesn't matter that it is a coefficient times an x; that is OK.0671
It is allowed to be a coefficient times some variable raised to some exponent.0674
But that is the whole thing--that is one of our terms for this.0678
A binomial has two terms; it could be x + 1 (as simple as that), or it could be more complex, like -52x^{7} + 892x.0681
Or we could have one term, which is x, or maybe even something really, really large, like x raised to the 1,845.0689
All right, the distributive property: very often, we are going to need to either factor polynomials--0697
break them into their multiplicative pieces--or expand these factors into a polynomial that is in general form.0702
So, take these multiplicative pieces, and then combine them together to get something larger0708
that gives us the whole polynomial in that general form that we saw of _x to the exponent + _x to the exponent.0713
We will see why this matters later on, especially in our next lesson, where we will talk about roots and zeroes of polynomials.0720
But for now, it is really important to understand how we get somewhere from (x + 1)(x + 2) into x^{2} + 3x + 2.0727
This is probably going to be a bit of a review for most of you; but it is good to understand why this is happening,0734
as opposed to just being able to do it mechanically.0738
So, let's look at what is making it up.0740
The thing this comes from is the distributive property, which says how multiplication interacts with parentheses.0742
If something multiplies against parentheses, it distributes to every term that is separated by addition or separated by subtraction.0750
For example, if we have a(b + c), then the a gets distributed onto the b, and the a gets distributed onto the c.0756
So, we get ab + ac; that is how distribution works.0765
How is that connecting to FOIL-ing things--how is it connected to different multiplicative factors for polynomials?0770
Well, our distributive property is a(b + c) becomes ab + ac.0776
From this property, we can use that on two different things in parentheses.0781
We can distribute parentheses onto other parentheses; and the most basic form with two binomials,0786
which is to say two things with two terms--we have the FOIL method.0791
For example, if we have (a + b)(x + y), we can think of (a + b) as just being a block.0795
So, like a is a block in our top example up here, we can think of (a + b) as being a block down here.0802
(a + b) goes onto x, and (a + b) goes onto y; so we get (a + b) times x and (a + b) times y.0810
Then, we turn right around, and we distribute in the other direction.0817
We take x, and we distribute that onto the a and onto the b; and we take y, and we distribute that onto the a, and distribute that onto the b.0820
And so, we get ax + bx, and then ay + by.0827
Now, what does FOIL mean? FOIL is a mnemonic to help us remember the order of multiplication: Firsts, Outers, Inners, Lasts.0831
Let's see how that comes to be; that end would be this way, where it is (a + b)(x + y).0840
We would do the firsts; we would do a and x (those are the first things); so we would get ax.0847
And then next, we would do the outers; a is on the outside, and y is on the outside (the outer part of our parentheses); we get ay + ay.0854
b times x would be our inners, the things on the inner part of the parentheses...b times x.0865
And then, b times y would be our lasts, because they are the last thing in each of our parentheses; plus b times y.0872
And we see that these two things are exactly the same thing; it is just reordered.0879
So, the distributive property and FOIL have the same thing going on here.0884
It is just a way of being able to say, "How is this going to multiply? How is it going to distribute onto the other thing?"0888
The way that we are making this FOIL method is two distributions, one after another.0893
But when we are actually using the distributive property to multiply out polynomial factors,0897
we probably want to think in terms of this first term, times the other terms inside,0901
and then the second terms times the other terms inside, and then the third term, and so on, and so forth, and so on.0906
This idea can expand into working on much longer parentheses than just two terms inside of it.0913
So, instead of just using binomials, we could have something like (x^{2} + 2x + 2)(3x^{2} - x).0919
So now, our first one has three terms, as opposed to just two.0925
But the same method still works: we can have x^{2} times 3x^{2}, and then x^{2} times -x.0928
Next, we will do 2x times 3x^{2}, and then 2x times -x.0941
And then finally, we will do 2 times 3x^{2}, and 2 times -x; great.0952
Each term in the parenthetical group multiplies all of the terms in the other parenthetical group.0964
We have x^{2} multiplying against 3x^{2}, and then multiplying against -x.0969
So, each term in the parenthetical group--one of the things in our parentheses--multiplies all of the terms in the other parenthetical group.0975
We start with factors, and we multiply them out; when we do that, it is called expanding.0982
What we just saw here is called expanding.0986
When expanding, we are normally expected to simplify.0989
I didn't simplify this one, because we don't really want to get into having to do that right now.0992
But we could simplify it pretty easily at this point.0997
We would multiply things out; we would get x^{2} times 3x^{2} (becomes 3x^{4}).0999
And then, we would do that with all of the other ones, and eventually we could add like terms together.1004
And we could simplify this into one of our general-form polynomials of _x to the exponent + _x to the exponent + _x to the exponent.1008
We can get it back into that general form.1016
Expanding is also sometimes called FOIL-ing; now, this is technically incorrect for larger factors,1019
because remember: FOIL is based off of that mnemonic: Firsts, Outers, Inners, Lasts.1024
So, that requires it to be 2 and 2 (two binomials put together).1028
But when people say this, we still know what they mean; FOIL-ing just means...it is another way of saying "expanding."1032
So, when somebody says "FOIL these polynomials" or "expand these polynomials," they are really getting across the same idea.1039
Use the distributive property; simplify it.1044
The reverse process, taking a polynomial and breaking it up into those multiplicative factors, is called factoring.1047
So, when we have this large, general-form polynomial, and we break it into those pieces,1053
like (x^{2} + 2x + 2) and then (3x^{2} - x), that is breaking it into the multiplicative factors; so we call it factoring.1058
The long-term behavior of a polynomial is determined by the term that has the largest exponent.1068
Other terms can have an effect; but their effect will become less and less noticeable as x approaches either positive or negative infinity.1073
Basically, as x goes very far in either direction (either to the right or to the left),1080
it is going to end up being the case that the polynomial will be controlled1085
by whichever exponent is largest--the term that has the largest exponent.1088
Why is this the case? Well, let's consider: if we have x, x^{2}, x^{3}, x^{4}, and x^{5},1092
and we plug in different values for x, when we plug in 1, they end up pretty much all being the same.1098
1, 1, 1, 1, 1...they are all exactly the same.1103
We get nothing but the same thing out of each of them.1106
But if we plug in something different, like 2, we start to see differences come up: 2, 4, 8, 16, 32.1108
Of course, the differences aren't very large yet; but as the numbers get larger and larger that we are plugging in,1114
5, 25, 125, 625, 3125...the difference between x^{2} and x^{5} is now 3100.1121
And if we just get up to x as 10 (plug in 10 for x), we get 10, 100, 1000, 10000, 100000...1133
massive differences between x^{5} and x^{2}, or x^{5} and x.1143
Even the difference between x^{4} and x^{5} is a difference of 90,000.1147
And we are only at x = 10; clearly, x^{5}...if we place all of these side-by-side...is going to be the massive winner.1155
It is going to have huge amounts of control; it is going to contribute so much more to what the value will end up being1162
than either x, x^{2}, x^{3}, or x^{4}.1167
None of those are going to be nearly as important as x^{5}.1170
So, as x becomes very big (positive or negative), the polynomial will be controlled by whichever term has the largest exponent.1173
The term that has the largest exponent--in this case, when we compared these 5, it would be x^{5}.1180
Whatever has the largest exponent is going to end up taking over.1187
Even if it has a really, really tiny coefficient in front, like 0.0001 times x^{5}, that will eventually get cracked.1190
As x^{5} becomes larger and larger and larger, and we plug in fairly large x, like, say, 10000,1198
it will be able to knock out that coefficient and still be more important than x^{4}, x^{3}, x^{2}, x.1205
So, the only thing that really matters is which one has the largest exponent.1210
Once you can figure out that, you know which one is going to be in control of the function at the extreme values of ±∞.1214
One other thing can have an effect, though.1222
The leading coefficient is very important, because it is going to be able to flip it.1224
So, the largest exponent is the term that determines things; the term with the largest exponent determines what will happen.1229
But the coefficient on that term will also matter.1236
If the coefficient is positive, it behaves normally; but if the coefficient is negative, it is going to flip the term.1239
What do I mean by that? Well, let's look at x^{2}.1245
x^{2} has a normal parabola arc like that; but if we have -x^{2}, it is going to flip it.1248
So, with x^{2}, we end up going up on the left and up on the right.1256
But with -x^{2}, we end up going down on the right and down on the left.1259
So, it is going to be down on both sides, because the negative is flipping it.1264
This leading term, whether it is a plus or a minus in front, is going to have control over what happens.1268
Either we are doing things the normal way, or we are going to flip to the opposite of that.1276
So, when a polynomial is in standard form (which is to say that the largest exponent is in the front), we call this the leading coefficient test.1281
By knowing what the leading coefficient is and the degree of the polynomial, we will be able to know what the long-term behavior is.1288
All you need to know to use the leading coefficient test is the degree of the polynomial and the sign of the leading coefficient,1294
which is going to be either plus or minus (or negative, technically).1302
We know what its long-term behavior will be like; we will see some pictures on the next one.1307
Long-term behavior--what do we mean by that? That is what happens as x gets very big--1310
as x goes out to plus or minus infinity, as it gets very, very far away.1315
We haven't really determined what it means by very, very far away; but it is just eventually, in the long run, how things will behave.1320
Let's look at some pictures to understand what this means.1328
So, for the leading coefficient test, if we have an even degree (which is a polynomial1330
where the leading exponent is going to be even, like x^{2}, x^{4}, x^{6}, x^{8}, etc.),1335
then if the coefficient is positive, on the right and on the left, we are going to be going up,1345
because, when we plug in a very large positive number, it is going to still stay a very large positive number.1351
If we plug in a very large negative number, then that even exponent will flip it to being positive; so we will still be going up.1357
On the other hand, if we have a coefficient that is negative, then when we plug in a very large one,1364
we will get a very large number out; but it will then get flipped to going negative.1369
If we plug in a very large negative number, then it will get flipped to positive.1372
But once again, the negative coefficient will hit it; and so it will go down.1376
So, for an even degree with a positive coefficient, both the left and the right side go up.1379
If we are an even degree with a negative coefficient, both the right and the left side go down.1383
An odd one, though (that is to say, something like x^{1}, x^{3}, x^{5}, x^{7},1389
and so on and so on)...if the coefficient is positive, then as we go very far to the right,1396
we are going to go up; we plug in a very large number, and we will get a very large positive number out of it.1402
But if we plug in a very large negative number, it has an odd exponent; so x^{3}...1407
-2 plugged into x^{3} is -2 times -2 times -2; three negative signs means we are left with a negative sign; so we would get -8.1415
So, it starts to go down as it goes negative and negative.1422
On the other hand, if we had a negative coefficient, then we would end up flipping that.1426
As we plug in very large positive numbers, they will get flipped down to going in the negative way.1432
And if we plug in a very large negative number, it will come out negative;1437
but then it will get flipped by that coefficient, and it will go positive; it will go up; great.1440
So, the leading coefficient test is: if we know it is an even and a positive, it is going to be up on both sides.1445
If it is an even, and it is a negative in front, then it is going to be down on both sides.1450
Odd and positive is going to be down on the left, up on the right.1453
And odd with a negative is going to be down on the right, up on the left.1458
So, just keep those pictures in mind, and think of flipping.1464
Now, notice that in the middle, we have these dashed lines; and what those dashed lines say1467
is that we don't have any idea what the middle part is going to look like.1472
The leading coefficient test only tells us what happens on the extremes--on the far left and the far right.1475
What is going to happen eventually, one day, in the long term?1484
But what happens in the middle--that is going to depend on the specific thing.1488
It could be very interesting; it could be not that interesting; we don't know what it is going to be until we get at specific function that we are looking at.1490
Then, we can figure out what it is going to be exactly.1501
The leading coefficient test just tells us what is going to happen in the long term, to the very far right and the very far left--those portions.1503
All right, we are ready for some examples.1513
What is n, the degree, for 2x^{4} - 8x^{3} + 2^{5}x - 19?1514
Remember, the degree is the largest exponent on a variable.1522
We go through; we look at all of our variables; and we see that this is the largest exponent on any of our variables.1532
We might notice this 2^{5}; but then we remember that it has to be a variable.1539
So, the 2^{5} doesn't get considered; and so, x^{4} is the case.1544
n is just our degree for a polynomial; so we have n = 4; and what is a_{n}?1549
Remember, the first one was an here; and then a_{3} goes with the x^{3}.1555
And then, a_{2} would go with x^{2}; but where is that?1561
First, a_{n} is 2, which is also the exact same thing as a_{4}, because we have n as 4, so a_{4} = 2.1565
What is a_{3}? Well, what is the coefficient for x^{3}? That is 8.1578
What is the coefficient for the x^{2}? We look at this, and we realize that that didn't show up at all.1585
But we could rewrite this as 2x^{4} - 8x^{3} + 0x^{2},1591
because x^{2} never showed up, so it must have been taken out by something; it has been taken out by this 0.1598
Plus 2^{5}x, minus 19...so if that is the case, then it must be that it is a_{2} = 0.1603
The plugging in for a_{2} must be 0, because it has to be able to take out that x^{2} term.1615
Then, from there, we just continue: a_{1} is equal to 2^{5};1621
and finally, our last one is a_{0} at the very end; a_{0} equals -19.1625
So now we see what all of the coefficients are; we know what the degree is; great.1633
The second example: Expand and simplify this expression; we have (x - 2)^{2}(x^{3} - x + 3).1637
The first thing we have to do is realize that (x - 2)^{2} is just the same thing as (x - 2)(x - 2).1644
If I have smiley face squared, then that is the same thing as smiley face times smiley face.1651
If I have (x - 2)^{2}, then that is just (x - 2)(x - 2).1657
Then, x^{3} - x + 3: let's start on the left and work our way to the right.1661
(x - 2)(x - 2); well, that will get us x^{2} (x times x) - 2x - 2x -2(-2) (becomes + 4).1667
And then, x^{3} - x + 3...I haven't really worked with that yet.1684
Let's simplify the left side first: x - 2x - 2x + 4...sorry, not x times x; x times x becomes x^{2}; sorry about that.1688
We have x^{2} - 2x - 2x + 4; x^{2} - 2x - 2x becomes x^{2} - 4x, as we combine like terms; + 4.1698
Then, times the quantity x^{3} - x + 3.1707
All right, let's use different colors for the various pieces we have here.1712
x^{2} times x^{3} becomes x^{5}; x^{2} times -x becomes -x^{3};1715
x^{2} times positive 3 becomes + 3x^{2}.1724
The next color is for -4x; that was our x^{2} portion.1728
-4x we will do in blue; so -4x times x^{3} will become -4x^{4}.1733
-4x times -x becomes positive 4x^{2}; and then, -4x times positive 3 becomes -12x.1741
The final one we will do in green; 4 times x^{3} becomes + 4x^{3};1751
4 times -x becomes -4x; 4 times 3 becomes + 12.1757
Great; now we have to simplify this.1763
Now, this isn't too difficult to simplify, but it is easy to get lost.1765
Each of the steps that we are about to do is pretty easy; the hard part is making sure we don't accidentally have any tiny missteps as we work through this.1769
So, I would recommend checking and doing them by exponent.1777
The first thing we will do is look at all the x^{5}'s.1780
We see that there are no other x^{5}'s, so we just bring it down; we have x^{5},1783
and then we will cross this out, so that we don't accidentally see it again, and don't accidentally end up trying to use it again.1787
Next, we have x^{4}'s; where are our x^{4}'s? We have -4x^{4}.1792
Do we have any other x^{4}'s? We look through it; no, we don't have any other x^{4}'s.1796
So, we bring that down; -4x^{4}; and then we cross it out, so we don't accidentally try to use it again.1801
Next, let's look for our x cubeds; we have an x cubed right here--anywhere else?--yes, we do; we have another x cubed here.1807
So, we bring those together: -x^{3} + 4x^{3} becomes + 3x^{3}.1812
-1 + 4...we get + 3x^{3}; and then we cross those out.1819
Next are 3x^{2} and 4x^{2}; there are no other x squareds; 3x^{2} + 4x^{2} becomes 7x^{2}.1825
We cross those out; next are x's; -12x - 4x; combine those together, and we get -16x.1833
Take those out; and + 12; there we are.1841
Now, you don't have to do this method of saying, "Here are my x^{5}'s; here are my x^{4}'s" and so on,1846
and so on, and then crossing them out as you go.1852
But this is a great way to make sure you don't accidentally make a mistake.1854
It is easy, when you are working with this many terms and trying to put them together and simplify,1856
to make one tiny mistake and lost the entire problem because of it.1861
So, it is a good idea to have some method of being able to follow your work and make sure1864
you don't accidentally try to do the same thing twice, or completely miss a term.1867
All right, the next one: Give an example of a quadratic trinomial, a cubic monomial, and a linear binomial.1871
Quadratic trinomial: remember, quadratic meant degree 2; and then, trinomial meant three terms.1879
A cubic monomial is a degree 3 (cubic means degree 3); monomial...mono- means single, like monorail,1890
a train track with one rail (not really a train anymore); a monomial is one term.1900
And then finally, linear is degree one; and binomial is two terms (bi- like bicycle); great.1907
So, if we want to give an example of this, we just need something that is degree 2 and 3 terms.1919
If it is degree 2 and it has 3 terms, then we are going to have something that has x^{2} at the front;1927
and it has to have blank spots for a total of three things.1932
Now, we can't have zeroes show up in these, because then it would disappear and we wouldn't have a term there.1938
We will have to put in something; so let's call it 5x^{2} + 3x, and we will make it -17.1942
You could plug anything into these blanks, and the answer would still be correct.1949
5x^{2} + 3x - 17; there is our quadratic trinomial.1952
Next, we do a cubic monomial; we know it has to be degree 3.1958
Degree 3 means it has to be x^{3}; and it is only one term, so there is going to be a blank in front of the x^{3}.1963
But we are not allowed to have any other blank things, because if we did, then we would have more than one term.1969
We are only allowed to have one term; so all of that gets taken right out--it disappears.1977
We have just _x^{3}; we plug whatever we feel like in...I feel like -47, so we get -47x^{3}.1982
Great; the final one--we have a linear binomial: a binomial has to have two terms, and linear is degree 1.1990
So, we have x^{1}, with some blank in front of it, plus blank, _x + _.1998
What goes in those blanks? Whatever we feel like.2004
We are not allowed to have any other blanks, though, because then we would have more than two terms.2006
Also, we can't have any more blanks, because we are linear, and that is the most that we have there.2009
So, _x + _...let's put in 1 for the x and -7 for the constant; so we have x - 7.2013
Great; the last thing--explain why it is impossible to have a linear trinomial.2022
So, if you are going to have a linear trinomial, let's see what that structure has to be.2027
Well, if we are linear, we know that x is going to be at the front.2032
And so, if we do the normal structure that we have for polynomials, it will be _x + _.2036
But if it is a trinomial, "trinomial" means we have to have three terms.2041
So, if we try to force on a third term, we would have to have _x^{2}.2046
We already have _x + _, so the only way to go is to go to the left; we have to have higher and higher exponents.2051
So, _x^{2}...all of a sudden, now we are a trinomial, but we are not linear anymore.2058
So, it means that we can't have both of these things at the same time.2065
We can't both be linear and have a third term; otherwise we would have to have x^{2},2068
at which point we wouldn't be linear anymore; we would be quadratic.2073
So, it is going to be one or the other; you can't be both a trinomial and a linear function.2076
The final example: What is the degree of y = (-2x^{2} + 4)^{407}?2082
Now, you see this at first, and you might get scared, because you think, "I can't possibly expand 407 times--I can't do that!"2088
But don't worry; all they asked for was the degree.2094
So notice: if I have (x^{2} + 3)(x^{5} + 48), do I have to look at anything else2097
to figure out what the degree is going to be, other than the front parts?2107
No, because I know only the x^{2} and the x^{5} are going to come together to make x^{7}.2110
And there is going to be other stuff; but I know I can't get any higher exponents out of this than the x^{7}.2114
It is going to be the leading term that will have the highest exponent.2120
It is going to be the exact same thing on this one.2123
It is going to be that -2x^{2}; it is a question of how many times -2x^{2} hits -2x^{2}.2126
That is the only thing that is going to be able to really bring increases of the degree.2132
There is going to be a whole bunch of other stuff; but we are not concerned with it, because all that they asked for was the degree.2136
It is going to be -2x^{2} raised to the 407, plus other stuff.2141
But we don't care about the other stuff: -2x^{2} to the 407...we distribute that...-2^{407}(x^{2})^{407}.2148
So, if we have 407 x^{2}, then it is x^{2} times x^{2} times x^{2} times x^{2}...2160
So, it is going to be the same thing as x^{2(407)}, because they are going to iterate that many times; it is going to hit that many times.2165
So, we have (-2)^{407}, times x^{2(407)}; (-2)^{407}x^{814}.2174
So, our degree is n = 814; that is our degree for this polynomial.2190
Now, as x goes very far to the left (x goes to -∞), will y go up or down (y approaches +∞ or y approaches -∞)?2198
And then, what about as x goes very far to the right--as x goes to positive infinity?2206
So, to do that, we need the leading coefficient test.2209
At this point, we already know what the degree of this polynomial is.2214
This polynomial is n = 814; so it is an even-degree polynomial.2219
Now, we want to figure out what our leading coefficient is; is it positive or negative--plus or minus?2226
We do that: -2 to the 407, times x to the 814...well, if it is a negative raised to an even number, they will all get canceled out.2233
If it is a negative raised to an odd number, one of them remains, because it will end up getting to stay around.2243
All of the even part will get canceled out, but that odd is an extra +1, so it stays around.2250
So, we will get -2^{407}x^{814}; that means we have a negative sign right here.2255
So, by the leading coefficient test, we have negative and even; negative and even means an even one.2265
Even normally goes in the same way that a parabola goes; it cups up, normally (even at positive).2276
But even at negative will flip that cupping shape, and we will get that.2283
Now of course, we don't actually know what is in the middle; all we know is the extremes,2287
because that is all we were guaranteed from the leading coefficient test.2292
But that is all we have to figure out, because it is as x approaches negative infinity.2295
So, from this, we see, even as it goes negative, that we go down on the left and down on the right.2298
So, as x approaches negative infinity--as x goes very far to the left--we are going to approach y going to negative infinity.2304
As x goes very far to the right (x goes to infinity), we are going to get y going to negative infinity, once again.2312
All right, great--the leading coefficient test should be able to figure that out.2321
All right, we will see you at Educator.com later.2324
Next time, we will look at roots and zeroes of polynomials and get a really good understanding of how these things are working.2326
All right--goodbye!2331
1 answer
Last reply by: Professor Selhorst-Jones
Mon Oct 20, 2014 11:32 AM
Post by Saadman Elman on October 18, 2014
[EDIT BY TEACHER: This student noticed a minor mistake in Example 1. Around 26:20, I say that a_3 = 8, but that is INCORRECT. It should actually be "-8". The negative sign is part of the number because of the minus sign. I'll try to get this fixed in the near future, but for now I'm just going to leave this note in case it confuses any other students.
Thanks for pointing this out to me.]
In example no. 1 a3 or a subscript 3 is -8 not positive 8. I would like this confirmed thanks. Your explanation was helpful!
1 answer
Last reply by: Professor Selhorst-Jones
Sun Sep 7, 2014 10:50 AM
Post by David Llewellyn on September 7, 2014
Is a0 + a-1.x^-1 + a-2.x^-2 + a-3.x^-3 + ..... + a-n.x^-n considered a polynomial and would you consider it to be of degree -1 or -n or even zero as a0 could be written as a0.x^0 and x^0 would then be the largest, in the sense of furthest to the right on the number line, exponent of x? I suspect that it is a polynomial of degree -1 but I'd like this confirmed.