Types of Algebraic Expressions: Monomials, Binomials and Polynomials Video Lecture - Class 8

Welcome back to Mr. Ace Math. This lesson
is on types of algebraic expressions. Some
basic stuff you should know before you begin
are the basic operations, how to add integers,
and how to subtract integers. So, make sure
you know these things, and then let's get
started. So first of all, what are algebraic
expressions? First of all, expressions. An
expression is a mathematical sentence. And
it does not have an equal sign. Be very careful.
A lot of people have heard the word equation.
And they think automatically expressions
and equations are the same thing. But guess
what? They're not! Expressions do not have
an equal sign at all. So what is an algebraic
expression? An algebraic expression is just
like an expression with a variable. So it
doesn't have just numerals like standard numerical
expressions. But it includes a variable. That's
what makes it algebraic. Think about it. In
algebra, you have variables. And terms. A
term is an item in an expression separated
by addition or subtraction. A lot of teachers
try to go on and on about different technicalities
and different ways to say it, but I honestly
think that saying that it's separated by addition
or subtraction is probably the most straight
forward, easiest way to see how many terms
there are in an expression. Now there are
4 types of algebraic expressions. The first
is a monomial. A monomial is an expression
with 1 term. The reason I highlighted the
part that says "mono" is because "mono" is
actually a prefix that means "one." So, a
monomial is an expression with one term. For
example, 3x. There's nothing here that's separated
by addition or subtraction. Therefore, there's
only 1 term. 3x. Another example. -7x squared.
Again, even though there's a negative there,
we're not adding or subtracting anything.
So it's still one term. It's still a monomial.
The next one. -23a to the 4th b to the fifth.
It looks like a lot because you have a number,
you have -23, you have one variable here,
and another variable here. But it's still
not separated by addition or subtraction.
Therefore, it's still only 1 term. Same thing
here. 8w x to the fourth times y. Again, there's
nothing separated by addition or subtraction,
so all of that, no matter how long it is,
it's still only 1 term. The second is a binomial.
A binomial is an expression with 2 terms.
The prefix "bi" means "two." So, it's an expression
with 2 terms. For example, we have 3x plus
2. Now, here we have things separated by addition
or subtraction. Here it's addition. So our
first term is 3x and our second term is plus
2. So, it's a binomial. Here, negative 7cd
minus x. We have 2 terms. The first is negative
7cd and the second is minus x or negative
x. Now you're probably... you're going to
hear me say a lot negative x or minus x or
negative 7 or minus 7. Keep in mind, algebraically,
negative and minus mean the same thing. So
they're going to be interchangable. So we
have 1 term here, negative 7cd and negative
x. Those are our 2 terms. Here, our first
term is w to the fourth y. Our second term
12. Here, our first term we have two very
long terms here. What's the first? Well, where
are we separating by addition or subtraction?
Well, we're separating by addition, and that
addition sign separates the first term here
negative 8 q squared rs and our second term,
which is plus or positive 3ef to the fifth
g. Now don't get intimidated by the fact that
there's a lot of variables here. The third
one is a trinomial. And a trinomial is an
expression with three terms. Again, it's pretty
straightforward. The prefix "tri" means "three."
The same thing as you hear in the word "triangle."
A triangle is a shape with three sides. A
tripod, if you have a tripod for a camera,
it's actually a stand that has 3 legs. "Tri"
always means "three." For example, we have
3x minus 8y plus 2. Now, remember, a term
is separated by addition or subtraction. Here,
we have minus 8y, OK, that separates these
2. So my first term is 3x and my second term
is minus 8y or negative 8y and my third term
here is plus 2 or positive 2. Here, my three
terms are x squared, minus 10x plus 8. All
three of those are terms. How about in this
example? There's a lot here. OK, we have negative
9ab squared plus 10 minus 8b to the third.
So, what do you think? Well, where are we
separating by addition or subtraction? Well
our first separation by addition is here.
So our first term is going to be negative
9ab squared. So where else are we separating?
Well we're separating by subtraction here,
so that separates this. So that's going to
be positive or plus 10, that's my second term.
And my third term is right here. Negative
8b to the third. How about this one? Take
a second. Think about it. Remember, we're
separating by addition or subtraction. So,
we're going to separate 10g to the third h
plus 3f minus 17f squared. So we're separating
this term and this term here by addition,
so our first term is 10g to the third h. Our
second term is plus 3f and our third term
is minus 17f squared or negative 17f squared.
We also have polynomials. Now "poly" is basically
just an expression with more than 1 term.
Binomials and trinomials are types of polynomials.
It's important that you keep that in mind.
Binomials and trinomials are polynomials,
and the reason why is because this word "poly,"
the word "poly" means "many." OK, so polynomials
really just means an expression with many
terms. Let's take a look. We have 3x plus
2. This is 3x and that's positive 2. There
are my two terms. The first term is 3x. The
second term is positive 2. And because it
has two terms it is a binomial, and because
it is a binomial, it is still technically
also a polynomial. And this example here.
3x minus 8y plus 2. Well how many terms do
I have? I have 3. I've got 3x negative 8y
and positive 2. OK there's three terms there.
Therefore it is a trinomial. And since it's
a trinomial it is technically also a polynomial.
So keep in mind. That's very important. A
lot of people think that when you say binomial
or trinomial or polynomial that they're separate
things. Granted, a binomial and a trinomial
are different. But a polynomial is more than
one, so binomials and trinomials are still
polynomials. Here we have 3xy. Now let's think
about that. Is that a monomial, a binomial,
a trinomial, a polynomial? Well, it's a monomial.
It is not a polynomial because it only has
one term. Remember, we said that a polynomial
is an expression with many terms. And those
terms will be set up separated by addition
or subtraction. Since there's nothing there
separated by addition or subtraction, it is
only a monomial. OK, so what kind of expression
is this? How many terms are there? Remember,
to determine the number of terms, we have
to see how many things are separated by addition
or subtraction. Well, these two terms are
separated by addition. These two are separated
by subtraction, and these two are separated
by addition. So, I've got my first term right
here, which is 4w. Then I'm adding this, so
positive 3x or plus 3x is my second term I'm
subtracting this, so negative 8y or minus
8y is my third term. And positive 2 or plus
2 is my fourth term. So I have 4 terms. Therefore,
this is a polynomial. So here's your "Pause
& Practice." Just pause and practice. When
you're done, un-pause the video. After a 3,
2, 1 countdown, your answers will be displayed.
Go! OK, so let's go over these answers. Number
1, the answer is monomial. Number 2 is a trinomial.
3 is a binomial. 4 is a polynomial. 5 is a
binomial. 6 is a monomial. And 7 is a polynomial.
Reflection. A mathematical sentence that does
not have an equal sign is called an expression.
Remember, there's a huge difference between
an expression and an equation. Expressions
do not have an equal sign. An equation does.
An expression with a variable is called an
algebraic expression. Remember, algebra involves
variables. Therefore, if you have an expression
with a variable, it's called an algebraic
expression. Each item in an expression that
is separated by addition or subtraction is
called a term. An algebraic expression with
only one term is a monomial. An algebraic
expression with two terms is a binomial. An
algebraic expression with three terms is a
trinomial. And an algebraic expression with
more than one term is a polynomial. And remember
what we said before. Binomials and trinomials
are types of polynomials because a polynomial
has more than one term. A binomial has more
than one term. A trinomial has more than one
term. Therefore, they are both polynomials.
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