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Studying for a test? Prepare with these 8 lessons on Module 4: Expressions and equations.

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# Terms, factors, & coefficients

Video transcript

What I want to do in this video
is think about how expressions are formed and the
words we use to describe the different parts
of an expression. And the reason
why this is useful is when you hear other people
refer to some expression and say, oh, I don't agree
with the second term, or the third term
has four factors, or why is the coefficient
on that term 6, you'll know what
they're talking about, and you can communicate
in the same way. So let's think about what
those words actually mean. So we have an expression here. And the first thing
I want to think about are the terms of an
expression or what a term is. And one way to think about it
is the terms are the things that are getting added
and subtracted. So, for example, in this
expression right over here, you have three things that are
getting added and subtracted. The first thing, you're
taking 2 times 3. You're adding that to 4. And then from that,
you're subtracting 7y. So in this example,
you have three terms. The first term is 2 times 3. The second term is
just the number 4. And the third term is 7 times y. Now, let's think about
the term "factor." And when people are
talking about a factor, especially in terms
of an expression, they're talking
about the things that are getting multiplied
in each term. So, for example,
if you said, what are the factors
of the first term? The first term refers to this
one right over here-- 2 times 3. And there's two factors. There's a 2 and
a 3, and they are being multiplied by each other. So here you have two
factors in the first term. What about the second term? This was the first term. The second term here has
only one factor, just the 4. It's not being
multiplied by anything. And the third term here,
once again, has two factors. It's the product of 7 times y. So we have two factors here. We have a 7 and a y. And this constant factor
here, this number 7 that is multiplying
the variable, also has a special name. It is called the coefficient
of this term-- coefficient. And the coefficient
is the nonvariable that multiplies the
rest of the term. That's one way of
thinking about it. So here's 7y. Even if it was 7xy
or 7xyz or 7xyz squared, that nonvariable that's
multiplying everything else, we would consider to
be the coefficient. Now, let's do a
few more examples. And in each of these--
I encourage you actually right now to pause the video--
think about what the terms are. How many terms are there
in each expression, how many factors in each term,
and what are the coefficients? So let's look at this first one. It's clear that we have three
things being added together. This is the first term. This is the second term. And this is the third term. So this is the first term. This is the second term. This is the third term. And they each have two factors. This first one has
the factors 3 and x. The second one here has
the factors x and y. And this third one has
the factors y and z. Now, what are the
coefficients here? Well, remember, coefficients
was a nonvariable multiplying a bunch
of other variables. And so here, the coefficient in
this first term right over here is a 3. Now, you might be
saying, well, what about the coefficients on
these terms right over here? Depending on how
you think about it, one way to say it
is, well, xy is the same thing as 1 times xy. So some people would
say that, hey, you have a coefficient
of 1 here on the xy. Or it's implicitly there. It wasn't written, but you're
multiplying everything by 1. And that might be
subject to a little bit of interpretation
one way or another. Now, this one is
really interesting, because if we look at
the bigger expression, if we look at the
whole thing, it's clearly made up of three terms. The first term is xyz. The second term is x plus
1, that whole thing times y. And then the third term is 4x. And if you look at that level,
if you look at the first term, and you say, well, how many
factors does that have? Well, you would say that it has
three factors-- x, y, and z. How many factors does
the second term have? Well, you could say,
well, it has two factors. One factor is x plus y, and
then the other factor is y. The first factor is x plus 1. And the second one is y. It's multiplying
this expression. This smaller expression
itself is one of the factors. And the other one is y. And then this third one also
has two factors, a 4 and an x. And if someone said, hey, what's
the coefficient on this term? You would say, hey, look,
the coefficient is the 4. Now, let's look at
this one over here. Actually, before I look at that
one, what was interesting about this is that here you had
a little smaller expression itself acting as
one of the factors. So then you can go
and then zoom in on this expression
right over here. And you can ask
the same question. On this smaller expression,
how many terms does it have? Well, it has two
terms-- an x and a 1. Those are the two things
being added or subtracted. And each of them have
exactly one factor. So when we're giving
these, you can keep nesting these expressions
to think about when you talk about terms or factors
or factors of terms, you have to really specify
what part of the nesting you're thinking about. If you're talking about the
terms of this whole expression, there's one, two, three. But then you could look at this
subexpression, which itself is a factor of a term,
and say, oh, well, there's only two
terms in this one. Now, let's look at this one. How many terms? Well, once again,
there's clearly three. Actually, let me add
one more, because I'm tired of expressions
with three terms. So I'm just going
to add a 1 here. So now, we clearly
have four terms. This is the first term, second
term, third term, fourth term. And how many factors
are in each of them? Well, this is interesting. You might say, well, the
factors are the things that are being multiplied. But here I'm dividing by y. Remember, dividing by
y is the same thing as multiplying by
its reciprocal. So it would usually
be considered to have three
factors here, where the factors are 3x and 1/y. If you multiplied 3
times x times 1/y, you're going to get exactly
what you have right over here. So you would say this
has three factors. If someone asks, what's
the coefficient here? Well, you'd say, well,
that 3 is the coefficient. Here how many
factors do you have? And this is a little bit
tricky, because you might say, well, isn't 5x squared times
y, isn't that equal to 5 times x times x times y? And you'd be right. So it would be very tempting to
say that you have four factors. But the convention, the
tradition that most people use, is that they
consider the exponent with x as a base as just
one factor, this as just one factor. So traditionally, people will
say this has three factors. It has a 5x squared
and a y. x squared is just considered a factor. And once again, what's
the coefficient? It's the 5. So with that in mind,
how many factors here? Well, you have
three factors here. You have an x. You have a y squared. And you have a z to the fifth. And then, finally, this last
term, it's a constant term. How many factors does it have? Well, it's just 1. It's just got a 1 sitting there. It's not being
multiplied by anything.