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Factoring higher-degree polynomials: Common factor

Sal factors 16x^3+24x^2+9x as (x)(4x+3)^2.

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  • blobby green style avatar for user Kevin Vernon
    Would x(-4x-3)(-4x-3)=x(-4x-3)^2 also be a solution to the problem it seems to work
    (16 votes)
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  • duskpin ultimate style avatar for user musiclover2721
    I don't get FOIL .. what is it?
    (8 votes)
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  • blobby green style avatar for user monkeypnut
    so at he is explaining the answer where does the 2 go? and by 2 I mean the one from 2 times 4 times 3x.
    (4 votes)
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  • leaf red style avatar for user Anirudh Kannan
    Ok, I need some help on this problem. 486 + 108x + 6x^2.

    (I have already figured out that a is 9 and b is x. But, what is the number that is outside of the parentheses?)
    (3 votes)
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    • mr pink green style avatar for user David Severin
      Since all are even, they are divisible by 2, since all digits of each add to a number divisible by 3 (18, 9, and 6) they are divisible by 6, so 6 can come out to yield 6( x^2 + 18x + 81), since 9*9 = 81 and 9+9 = 18, you do in fact get 6(x+9)^2.
      (3 votes)
  • blobby green style avatar for user skhan
    how do I factor xy(x-2y)+3y(2y-x)^2?
    (0 votes)
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    • stelly blue style avatar for user Kim Seidel
      There are 2 ways:

      1) Completely simplify the polynomial, then factor
      Simplifying: xy(x-2y)+3y(2y-x)^2
      = x^2y-2xy^2 + 3y(4y^2-4xy+x^2)
      = x^2y-2xy^2+12y^3-12xy^2+3x^2y
      = 4x^2y-14xy^2+12y^3
      Factoring...
      -- Factor out GCF=2y: 2y [2x^2-7xy+6y^2]
      -- Factor trinomial using grouping. AC=12
      Find factors of 12 that add to -7. Use -4 and -3
      Use these to expand middle term
      2y [2x^2-4xy-3xy+6y^2]
      = 2y [2x(x-2y)-3y(x-2y)]
      = 2y (x-2y) (2x-3y)

      2) Use the structure of the polynomial to factor.
      This requires that you realize that (x-2y) = -1(2y-x) or (2y-x) = -1(x-2y). I'm going to use the 2nd version.
      If (2y-x) = -1(x-2y), then (2y-x)^2 = (-1)^2(x-2y)^2 = 1(x-2y)^2 = (x-2y)^2
      Using this, we can change the polynomial into:
      xy(x-2y)+3y(x-2y)^2
      This has 2 terms with a common factor of (x-2y). Use the distributive property to factor it out.
      xy(x-2y)+3y(x-2y)^2= (x-2y)[xy+3y(x-2y)]
      Simplify the 2nd factor:
      (x-2y)[xy+3y(x-2y)] = (x-2y)[xy+3xy-6y^2]
      = (x-2y)[4xy-6y^2]
      Then, factor out a GCF=2y from 2nd factor:
      (x-2y)[4xy-6y^2] = 2y (x-2y)(2x-3y)

      Hope this helps.
      (9 votes)
  • aqualine ultimate style avatar for user Ŧг๏รՇ ๏ฬɭ
    the way he does it is different that my teacher taught me is there more than one way to do this correctly?
    (3 votes)
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  • blobby green style avatar for user lydia Rose
    If I get the same factors but have them in a different order, it gets counted as wrong. Does it really matter what order they are in?
    For example
    Isn't (x-2)(x-1) the same thing as (x-1)(x-2)?
    (2 votes)
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  • blobby green style avatar for user brittany meeks
    At , how did he get x((4x)2
    (4 votes)
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  • male robot johnny style avatar for user manley90
    Would this equal 4x3+6x+14x2+21
    2x2(2x+7)+3(2x+7)?
    (2 votes)
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  • aqualine seed style avatar for user Deepthi Reddy
    So, this method must be used whenever we have a cubic expression, right?
    (1 vote)
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Video transcript

- [Voiceover] So let's say that we've got the polynomial 16x to the third plus 24x squared plus nine x. Now what I'd like you to do is pause the video and see if you could factor this polynomial completely. Now let's work through it together. So the first thing that you might notice is that all of the terms are divisible by x so we can actually factor out an x. So let's do that, and actually, if we look at these coefficients, it looks like, let's see, it looks like they don't have any common factors other than one. So it looks like the largest monomial that we can factor out is just going to be an x. So let's do that. Let's factor out an x. So then this is gonna be x times. When you factor out an x from 16x to the third you're gonna be left with 16x squared, and then plus 24x and then plus nine. Now this is starting to look interesting so let me just rewrite it. It's gonna be x times. This part over here looks interesting because when I see the 16x squared, this looks like a perfect square. Let me write it out. 16x squared. That's the same thing as four x squared, and then we have a nine over there, which is clearly a perfect square. That is three squared. Three squared. And when we look at this 24x, we see that it is four times three times two, and so we can write it as, let me write it this way. So this is going to be plus two times four times three x. So let me make it, so two times four times three times three x. Now why did I take the trouble, why did I take the trouble of writing everything like this? Because we see that it fits the pattern for a perfect square. What do I mean by that? Well in previous videos, we saw that if you have something of the form Ax plus B, and you were to square it, you're going to get Ax squared plus two ABx plus B squared, and we have that form right over here. This is the Ax squared. Let me do the same color. The Ax squared. Ax squared. We have the B squared. You have the B squared. And then you have the two ABx. Two ABx right over there. So this section, this entire section, we can rewrite as being we know it, what A and B are. A is four and B is three so this is going to be Ax, so four x plus four x plus b, which we know to be three. That whole thing, that whole thing squared, and now we can't forget this x out front so we have that x out front. And we're done. We have just factored this. We have just factored this completely. We could write it as x times four x plus three and then write out and then say times four x plus three or we could just write x times the quantity four x plus three squared. And so we've just factored it completely, and the key realizations here is well, one: what could I factor out from all of these terms? I could factor out an x from all of those terms, and then to realize what we had left over was a perfect square, really using this pattern that we were able to see in previous videos.