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## Multiplying monomials by polynomials

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# Multiplying monomials

CCSS.Math: ,

## Video transcript

- [Instructor] Let's say
that we wanted to multiply five x squared and,
I'll do this in purple, three x to the fifth,
what would this equal? Pause this video and see if you can reason through that a little bit. All right, now let's work
through this together. And really, all we're going to do is use properties of multiplication and use properties of exponents to essentially rewrite this expression. So we can just view this, if we're just multiplying
a bunch of things, it doesn't matter what
order we multiply them in. So you can just view this
as five times x squared times three times x to the fifth, or we could multiply our
five and three first, so you could view this as
five times three, times three, times x squared, times x squared, times x to the fifth,
times x to the fifth. And now what is five times three? I think you know that, that is 15. Now what is x squared
times x to the fifth? Now some of you might recognize that exponent properties
would come into play here. If I'm multiplying two things like this, so we have the some base
and different exponents, that this is going to
be equal to x to the, and we add these two exponents, x to the two plus five power,
or x to the seventh power. If what I just did seems
counterintuitive to you I'll just remind you, what is x squared? x squared is x times x. And what is x to the fifth? That is x times x times x times x times x. And if you multiply them all
together what do you get? Well you got seven x's and you multiply them all together and that is x to the seventh. And so there you have it, five x squared times three x to the fifth is 15x to the seventh power. So the key is, is look
at these coefficients, look at these numbers, a five
and a three, multiply those, and then for any variable you have, you have x here, so
you have a common base, then you can add those exponents, and what we just did is known
as multiplying monomials, which sounds very fancy, but
this is a monomial, monomial, and in the future we'll do multiplying things like polynomials where we have multiple of
these things added together. But that's all it is,
multiplying monomials. Let's do one more example, and let's use a different
variable this time, just to get some variety in there. Let's say we wanna multiply the monomial three t to the seventh power, times another monomial negative four t. Pause this video and see if
you can work through that. All right, so I'm gonna do
this one a little bit faster. I am going to look at the
three and the negative four and I'm gonna multiply those first, and I'm going to get a negative 12. And then if I were to want to multiply the t to the seventh times t, once again they're both
the variable t as our base, so that's going to be t to the seventh times t to the first
power, that's what t is, that's going to be t to
the seven plus one power, or t to the eighth. But there you go, we are done again, we just multiplied
another set of monomials.