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Factoring difference of squares: missing values

Sal analyzes the factorization of 3y^3-100y as 4y(My+g)(My-g) to find the possible values for the missing coefficient g.

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  • male robot donald style avatar for user Ella
    Can someone explain the rules of negatives while squaring numbers?
    (3 votes)
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    • aqualine ultimate style avatar for user Aubrey Janssen
      Remember that when you square a number you are multiplying it by itself. For example -5 squared would be -5*-5. and when you multiply two negative numbers you always get a positive, so no matter what number you SQUARE it will be positive. However, its different with cubing a number. -5 cubed is -5*-5*-5. you would first multiply the first two -5s and get 25, then you would multiply 25 by -5. And because you are multiplying a negative and a positive you will get a negative number, in this case -125. But now if you were to multiply -5 to the power of 4 you would be adding another -5 to the equation, making the answer positive again(-125*-5=626) See the pattern? Out of 5 to the power of 2,3,4..... every other one will be positive, or in other words the EVEN exponents yield POSITIVE answers and the ODD exponents yield NEGATIVE answers. This applies all numbers, not just 5 I encourage you to play around with your calculator to become more familiar with the concept. Just remember though that with most calculators, when adding exponents to a negative number you should put it in parenthesis with the exponent outside.
      (14 votes)
  • male robot hal style avatar for user Jonathan Markus
    couldnt g be equal to ten if you only factored out a 2y instead of 4y to begin with?
    (4 votes)
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    • winston baby style avatar for user Devon Horak
      If you only factored out a 2y, then you wouldn't be left with a difference of squares. You would be left with 2y(18y² - 50). Which would further factor down into 2y(6y + 10)(3y-5). The question specifically stated, however, they wanted it in the form 4y(My + g)(My - g), where M and g are both integers. Therefore, M has to be one integer, and b has to be a different integer. 2y(6y + 10)(3y - 5) does not satisfy these conditions. The first thing that jumps out at me would be that it isn't 4y outside of the parentheses, which they specifically asked for.
      You could take 2y(6y + 10)(3y - 5) and undistribute two out of the first parentheses though, which is another way that you could arrive at the right answer. Since (6y + 10) is the same as 2(3y + 5). But here, you notice that g is no longer equal to 10, and that was your question. And we are left with the same answer Sal got, which is 4y(3y + 5)(3y - 5).

      In order for g to have been equal to 10, the original equation needed to be 36y³ - 400y.
      4y(9y² - 100)
      4y(3y + 10)(3y - 10)
      (6 votes)
  • male robot hal style avatar for user DrumletNation
    And, you can do the same with cubes, right?
    (2 votes)
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    • mr pink green style avatar for user David Severin
      You can do differences of cubes as well as sum of cubes.
      x^3 - 1 = (x-1)(x^2 + x +1) if you expand, x^3 + x^2 + x - x^2 - x - 1, all middle terms cancel
      Similarly x^3 + 1 = (x + 1)(x^2 - 1x + 1) if you expand x^3 - x^2 + x + x^2 - x +1.
      You can add a coefficient to x^3 or a constant that are perfect cubes (1, 8, 27, 64, etc.)
      (3 votes)
  • male robot donald style avatar for user Ella
    In this video, does 4y apply to both parentheses or just one?
    (2 votes)
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    • blobby green style avatar for user stelbaums
      The 4y applies to just one of the parentheses in the final factored form. If you multiply the parentheses that you left alone with the product of 4y and the parentheses that you decided to multiply with it, you will see that this will lead back to the original polynomial.

      I had the same question when he extracted (or "undistributed") a -1 to get rid of the minus in a polynomial with a leading minus in previous videos, when I had factored it to 2 binomials I'm supposed to stick it back in front of them giving me -1 (thing1) ( thing2) and I would arrive back at the original polynomial only if I multiplied (thing1) with the -1 and then the product of that with (thing2), multiplying both of them with -1 wouldn't work.
      (2 votes)
  • blobby green style avatar for user anthonybui48
    how come "9y^2" is equaled to "3y^2"?
    (0 votes)
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  • leaf green style avatar for user Ryan Hettenbach
    Can you factor out one of the "y"s to make it?
    y(36y^2 - 100)
    Or can you not factor out a variable?
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      You can factor out the "y". In fact, that is the best place to start with factoring the polynomial you have. However, you aren't yet done. Your new binomial can be factored further.
      1) You can also pull out a common factor of 4 along with the "y": 4y (9y^2 - 25)
      2) Next, the binomial can be factored into 2 new binomials because it is a difference of 2 squares. The factors become: 4y (3y - 5)(3y + 5).

      Hope this helps.
      (1 vote)
  • blobby green style avatar for user Bob Donegan
    cant you just factor out a y and then you get y(36y^2-100)

    -then you have a difference of squares of y(6^2-10^2) => y(6+10)(6-10)

    You end up getting 10 as your G
    (1 vote)
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  • male robot hal style avatar for user Jonathan Markus
    couldnt g be equal to ten if you only factored out a 2y instead of 4y to begin with?
    (1 vote)
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  • blobby green style avatar for user AT
    I don't understand why Brandon's answer is incorrect.
    I tried: y(6y+10)(6y-10), which seemed to work.

    "Brandon wrote that g could be equal to 10," not that 10 was the only possible solution. Am I missing something?
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      The problem starts with 36y^3-100y.
      It tells you that the expression can be factored into:
      4y(My+g)(My-g)
      Notice, the factoring has already factored out 4y. If you factor out 4y, then you have 4y(9y^2-25)
      10 is not a factor of 25 which is why Brandon's answer doesn't work. The only way Brandon's answer works if if the 4y has not been factored out. But the factored form shows that it has been.

      Hope this helps.
      (1 vote)
  • male robot donald style avatar for user Ella
    What happenes when a value doesn't have anything to square like y itself?
    (1 vote)
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Video transcript

- [Voiceover] The polynomial expression, 36 y to the third minus 100y can be factored as 4y times all of this, My plus g, times My minus g, where M and g are integers. Sally wrote that g could be equal to 3. Brandon wrote that g could be equal to 10. Which student is correct? Now, when you look at this, it seems really daunting, all this M's and g's here, but we just need to realize that they're factoring out, first they factored out a 4y from the 36y to the third minus 100y, and it looks like whatever's left is a difference of squares, which they then factor even further. So I encourage you to pause the video and just factor this out as much as you can. First factoring out a 4y, and then we can think about what g is going to be equal to, or whether Sally or Brandon is correct. So, then now let's work through this together. So, if we look at, if we look at this expression right over here, we want to factor out a 4y, so 36y to the third minus 100y, that's the same thing as, 36y to the third is the same thing as 4y times, let's see, 4y times 9y squared, right? Because 4 times 9 is 36, and y times y squared is y to the third, so all I did to get the 9y squared is I divided 36 by 4 to get the 9, and I divided y to the third by the y to get y squared, so if you factor out a 4y, you're left with 9y squared for that first term, and then, for the second term, let's see, if we, we're going to subtract, if we factor out a 4y again, what's left over? 100 divided by 4 is 25, and then y divided by y is just 1, so we're left with a 25 here. So just to be clear what's going on, this 36y to the third, I just rewrote it as 4y times 9y squared. One way to think about it is I wrote it with the 4y factored out, and then, the 100y, right over here, I wrote it with the 4y factored out, so it's 4y times 25, and now it's very clear that we can factor out 4y from this entire thing, so we can factor out, you can think of it as undistributing the 4y, so this is going to be equal to 4y and what is left over? Well, if you factor out a 4y out of this first term, you're going to have a 9y squared, 9y squared, and then minus 25, and then we're going to be left with minus 25, and when we write it like this, we see what we have in parentheses here. This is a difference of squares, and, we could skip a step, but let me just rewrite it, so we could rewrite it as, literally, a difference of squares. 9y squared, that is the same thing as, that is the same thing as 3y, that whole thing to the 2nd power, 3 squared is 9, y squared is y squared, and then we have minus 25, we can rewrite as 5 squared, so you see, we have a difference of squares, and we've seen this pattern multiple times. If this is the first time you're seeing it, I encourage you to watch the videos on Khan Academy on difference of squares, but we know anything of the form, anything of the form a squared minus b squared, let me do it in that color, minus b squared can be factored as being equal to, this is equal to, if I were to write it as a product of two binomials, this is going to be equal to a plus b times a minus b, and you can verify that that works, if you've never seen this before, or you can watch those videos for review. So this right over here can be rewritten as 4y, which we factored out at the beginning, it's going to be times the product of two binomials for this part right over here, and so in this case, a is 3y, so it's going to be 3y plus 5, times 3y minus 5, so let me write that down. So 3y plus 5 times 3y minus 5. 3y minus 5. So now that we factored this, let's go back to what they originally told us, so then we have 4y, so this 4y corresponds to that 4y, right over there, and then you have My plus g, and then you have My minus g, so you could view the My, the My's right over there, that's the 3y right over there, so we could say that M is equal to 3, M is equal to 3, and then we do plus 5 and minus 5, plus g and minus g, so, g, if we're pattern matching right over here, g is going to be equal to 5. So g is equal to 5. So what's interesting about this problem is that neither one of them are correct, so, I could write, neither is correct. G is equal to 5. That was a tricky one.