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Multiplying monomials

A monomial is an expression of the form k⋅xⁿ, where k is a real number and n is a positive integer. It's basically a polynomial with a single term. When were are multiplying two monomials, we can rewrite the product as a single monomial using properties of multiplication and exponents.

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  • blobby green style avatar for user Adam H.
    At , I would like to make sure that I have the correct answer for the "cliffhanger". Is it 20x^9
    Thanks
    (37 votes)
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  • hopper cool style avatar for user Bob The Zealot
    Why are the exponents not multiplied?
    (13 votes)
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  • piceratops ultimate style avatar for user Rohan SP
    At , The answer for the problem 5x^4 * 4x^6 should be 20x^9 not 9x^18.
    (3 votes)
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    • stelly blue style avatar for user Kim Seidel
      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
      (7 votes)
  • leaf blue style avatar for user mileswilson000
    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?
    (4 votes)
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  • aqualine tree style avatar for user Emily Richmond
    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)
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    • female robot grace style avatar for user loumast17
      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
      (2 votes)
  • aqualine sapling style avatar for user Nigel Gunawardane
    People are commenting about timestamps at , but my video is only 3min and 15 sec?
    (3 votes)
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  • primosaur ultimate style avatar for user winslow.trullinger
    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)
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  • duskpin ultimate style avatar for user 23more_vaidehi
    At , for the cliffhanger....is the right answer 20x^9?
    (1 vote)
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  • blobby green style avatar for user #MentalAbuseToHumans
    I know how to do this part but I have an example on my homework like this:
    -3x(-2x^2 - 4x + 2) + 5 (x^2 - 12x) and I have no idea how to do it.
    (2 votes)
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  • old spice man blue style avatar for user angper23
    what if there was a negative in the equation ex:
    (y^8)(-2y^2
    (2 votes)
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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.