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### Course: Integrated math 3>Unit 1

Lesson 4: Multiplying monomials by polynomials

# Multiplying monomials

Learn how to multiply monomials like a pro! Discover how to multiply numbers and variables separately, then combine them for the final answer. Explore how exponent properties come into play when multiplying variables. Dive into examples with different variables for variety.

## Want to join the conversation?

• At , I would like to make sure that I have the correct answer for the "cliffhanger". Is it 20x^9
Thanks
(45 votes)
• Yes, you have the correct answer. Good job. Looks like they cut the video off too quickly.
(31 votes)
• Why are the exponents not multiplied?
(14 votes)
• x^a * x^b = x^a+b
If you have 2^2 * 2^3 the correct answer is 32. If you multiplied the exponents you would get 64. Hope that explains it.
(27 votes)
• If you did something like 5x^7 * 5x^4 you would get 25x^11. However, in the video he did 5^2 * 5^4 and got 5^6. Why did he not multiply the co-efficients?
(5 votes)
• Because the coefficient in the latter case is 1. Imagine multiplying x^2*x^4, then substitute x=5
(6 votes)
• At , The answer for the problem 5x^4 * 4x^6 should be 20x^9 not 9x^18.
(3 votes)
• At , Sal clearly tells you that he is going to show you a wrong answer. He is showing common errors that students make to try and help you avoid them.

And, FYI... your version: 5x^4 * 4x^6 = 20x^10, not 20x^9
Sal's problem: 5x^3 * 4x^6 does equal 20x^9
(9 votes)
• People are commenting about timestamps at , but my video is only 3min and 15 sec?
(4 votes)
• Yes... this video was more than 3 min... but now they had decided to change it to make shorter video to make students to understand it...
P.S. Click Top Voted and look at @Kim Seidel what he said.
(2 votes)
• ok so in my math class we are multiplying, dividing, adding, and subtracting monomials. but exactly how do we find out what the answer will be if all the problems are related to polynomials? what even are polynomials?
(3 votes)
• polynomials are a series of monomials that are added or subtracted together. There are a few extra rules though. The powers of the variables must be positive whole numbers, so no negative powers and no fractions or decimals as powers. Also coefficients can be fractons or decimals, but if there is a variable in a denominator then it is not a polynomial any more.

Technically monomials, binomials and trinomials are all kinds of polynomials. Also, if you have two terms in a polynomial that have the same variable to the same power, you need to combine it. so 5x^2 + 3x^2 would be 8x^2

As for how to solve them that is almost half a school year's worth of explanation. it will all be addressed on the site though.

One concept that is invaluable is this. All polynomials can be written as a series of binomials multiplied together. so something like (x+2)(x-1) and so on. You will learn how to write them like this. But you really want to know how to solve them when they are like this.

you will almost always have them equal to 0. so (x+2)(x-1) = 0

When you have this what happens if x was -2 or 1? well let's see.

if x was -2 then (x+2)(x-1) = (-2+2)(-2-1) = 0*-3 = 0.

Similarly if x = 1 then (x+2)(x-1) = 3*0 = 0.

So when you have the binomials multiplied together you just need to remember that the answers are when each binomial is 0.

ANother quick example. (x+2)(x-3)(x-5)(x-7) = 0 would have the answers -2, 3, 5 and 7.

You should learn how to do all this as you go, but consider this a head start
(3 votes)
• is it me, or do all his 5s look like Ss?
(3 votes)
• they just happen to look similar. Each person has their own handwriting so each to their own I guess :)
(3 votes)
• I understand that multiplying the same base the base does not change, but what if one of the bases is positive and the other negative? i.e. a base of -7 and 7. Does it become -49?
(2 votes)
• Exponents represent multiplication of a common base value. If the bases are not the same, then exponent properties don't apply. In certain cases, you may be able to clean up the signs to force a common base. For example: (-7)^2 = 7^2. So, if you had (-7)^2 * 7^3, you can simplify this to: 7^2 * 7^3 = 7^5

Does this answer you quesiton?
(4 votes)
• what is negative 9n to the 9 power times n to the 3 power
(2 votes)
• Use the properties of exponents. If we multiply 2 items with a common base, we add the exponents. For example: 2^3 * 2^5 = 2^(3+5) = 2^8

Try applying this to your problem. Comment back if you have questions.
(4 votes)
• What the student did wrong is they added the 5 and the 4 instead of multiplying them and then multiplied the 3 and 6 instead of adding.
(3 votes)

## 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.