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Let's talk about polynomials in this lecture.
When we looked at integers like, say, 324 in the expanded form,
we wrote it in this format.
We said 3 times 10 squared plus 2 times 10 plus 4 times 10
to the 0.
And we said that the numbers 3, 2, and 4
are called the coefficients.
3 is the coefficient of 100, or 10 squared.
2 is the coefficient of 10 to the 1,
and 4 is the coefficient of 10 to the 0.
We said before how, if you replace the base 10 with x's,
you end up with examples of polynomials.
So replacing base 10 with x, that's
considered playing, right?
Because we already have something,
and we want to replace pieces of it to see what
will happen, just for fun.
So 10 is replaced with x's, and we
have an example of a polynomial.
3x squared plus 2x plus 4.
To have any generic polynomial in one variable,
you can also change the coefficients.
We're not restricted to just integers.
You can replace the 3, 2, and 4 with real numbers,
and then you have a different polynomial.
So, again, we have 3.4x squared plus square of 2x plus 1/4.
That's an example of a polynomial.
The highest exponent on the variable
is called a degree of the polynomial in one variable.
So the degree of the polynomial in one variable
is the highest power that occurs in all of the terms
that you see here added together.
So let's take a look at some definitions.
A monomial is a product of real numbers and variables
to whole number exponents.
The real numbers are called the coefficient of the monomial,
and the sum of the exponents in the variable terms
is called the degree of the monomial.
So let's take some examples.
Here we have square root 2, x to the 100.
So the degree of that polynomial is 100.
The degree of this monomial is 5 because that's
the power of exponent.
If you have a term with multiple variables,
then to compute its degree, you look
at all the exponents involved on the variables
and take their sum.
So here, the degree of this monomial
is going to be 8, because 3 plus 5.
Those terms that you see, square root 2, negative 5.34,
negative 2/2345 are all coefficients of the monomial.
So now let's take a look at polynomial.
A polynomial is a finite sum of monomials.
That is, addition of one or more monomials.
And so the degree of a polynomial
is the highest degree across all the monomials
in the polynomial.
So here we have a polynomial, and you
can see that the degree of the polynomial
is 100, because if you look at each term,
100 is the highest exponent in the terms that are being added.
The number square root 2, 5, negative 4,
15 are called coefficients of the polynomial.
If you look at an example of our multivariable polynomial,
then we have to find the degree by looking
at all of the monomial terms, looking at its degree.
So, for example, at the monomial here,
what do you think its degree is?
Why don't you think about it?
So that degree is 12, because 10 plus 2.
The degree here is going to be 2 plus 4, 6.
Here's going to be 1 plus 1, 2.
So the highest degree monomial is this one,
and so the degree of this polynomial is going to be 12.
The coefficients are negative 34, a 5, and a negative 4.1.
Those are the coefficients of that polynomial.
You could also have polynomials with complex numbers
for coefficients.
So you're not just restricted to real numbers.
So you can pretty much have whatever
you want for coefficients, and the degree still is the same.
Addition of all the exponent terms
in each monomial, and then the highest power.
So why don't you see if you can find
the degree of this polynomial?
So here we have 1 and 3, so 4.
7, and here we have 7 plus 3, 10.
So this is the highest degree monomial,
so the degree of this polynomial is going to be 10.
An algebraic term is a product of numbers and variables
using any algebraic operations.
So you can have addition, subtraction,
multiplication, division.
You can also have any kind of exponents.
You are not restricted to just whole number exponents
like you are in polynomials and monomials.
So here's some examples.
This is an example of an algebraic term
because we have a numerator, and we have a denominator.
Such a term is actually called a rational expression.
It's not a polynomial because the variable
appears in the denominator.
And here are two examples of algebraic expressions
that are not polynomials.
You can see here we have a variable term
in the denominator.
Here we have a integer exponent, a negative exponent,
and that's not allowed for polynomials,
because you need whole number exponent.
OK, how about this one?
This is a polynomial because you have coefficients,
and you have variables raised to whole number powers,
so that one is an algebraic expression that's
also a polynomial.
So a polynomial is a special case of algebraic expression.
So let's just do some practice problems.
Pause the video.
Do the problems on your own so you
can see if you're getting it.
So is the first one a polynomial or not?
Going ahead and think.
Pause.
Assuming you have paused and come back,
here is the first answer.
First one is not a polynomial because you have square root
x, which is the same as x raised to a half power, which
is not a whole number power.
All right.
How about this one?
No, because of x squared. x squared is in the denominator.
Can't have variable terms in the denominator.
Same reason applies here.
You must be able to reason out why something is a polynomial
and why something is not a polynomial.
Here it's not a polynomial because this x is an exponent.
Remember, you can only have whole number exponents.
So it's an algebraic expression but not polynomial.
Homework.
See what you can do.
So in general, there are times when
we would like to refer to generic polynomials that
represent all types of polynomial.
So let's just start with a degree two
polynomial and one variable.
So 2x squared plus 3x plus 5.
However, what if I wanted to think about all degree
two polynomials in the second degree?
Then I would have to write it like this.
So we would have ax squared plus bx plus c
as a polynomial of degree two. a, b,
and c are real numbers, so this is how any degree two
polynomial would look like.
So you can see the problem now that we're going to run into.
If you have more than 26 terms, then what?
You're going to run out of letters.
So to fix the problem, mathematicians
have come up with an ingenious solution.
Look.
We can start with powers of x's here,
so we can have a constant term, which
is when there is no x term.
Then we have x to power 1 term, x to power 2 term, and so on.
The dot dot dot means you just continue like that.
How many of our terms do you have?
The generic coefficient, then, is represented
by variable a with a subscript n.
So a sub n is a number that would
be the coefficient of x to the power n.
So for example, a sub 2 is a number
that will be sitting in front of the x squared term.
a sub 1 is the number that sits in front of the x term.
So basically, we create infinitely many variables now.
We'll never run out of variables,
because we know there are infinitely many counting
numbers.
So this is a great way to represent
any generic polynomial of degree n.
a sub k is referred to as the coefficient of x to the k.
Remember, coefficient just means that it's
a number, like 4 or 4.5 or square root 2.
It's a real number.
So for example, if we have 9x to the fifth minus 3x
to the fourth plus 2, in this format,
all that means is that a sub 5, which
is the coefficient of degree 5, is 9.
Coefficient of degree 4 is negative 3,
and then the coefficients of third degree,
second degree, first degree are all 0,
so that's why they're missing.
And then the constant term a sub 0
is 2, which is the last term with no x in it.
That's referred to as a constant term.
The use of subscripts on coefficients
is a clever way to not run out of symbols.
So polynomials are used in physics, chemistry, business,
lots of other disciplines, and the polynomials
that we use frequently are given names
so you get familiar with these names, which
will help you later to recognize them.