If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Factoring two-variable quadratics: grouping

Sal factors 5rs+25r-3s-15 as (s+5)(5r-3). Created by Sal Khan and Monterey Institute for Technology and Education.

Want to join the conversation?

  • leaf green style avatar for user Derrick Malone
    Can you use FOIL to do this problem?
    (29 votes)
    Default Khan Academy avatar avatar for user
  • starky ultimate style avatar for user jackiexubm1
    I dont get this extra credit question my teacher gave me, and i am not sure if sal have videos about this type of question:
    a^2 x^2 + 4ab +b^2
    thank you very much!
    (6 votes)
    Default Khan Academy avatar avatar for user
  • leaf blue style avatar for user Michael Christian
    Doesn't FOIL stand for: First, Outside, Inner, and Last?
    (5 votes)
    Default Khan Academy avatar avatar for user
    • hopper cool style avatar for user Stuart Wickard
      Yes it does. Basically what that means is that when multiply to polynomials. In the case on (x+1)(y+2), you multiply the first (x and y), the outer (x and 2), the inner (1 and y), and the last (1 and 2). In the end you get x * y + x * 2 + 1 * y + 1 * 2. This can be simplified to xy+2x+y+2.
      (7 votes)
  • leaf red style avatar for user Sreeja J
    how can you do the with variables which have at least variable in common. for ex; 25x-5x-15xy-3y
    how can i factor 15 and 3? I know they're both divisible by 3 but, i don't know where to put the x.
    (5 votes)
    Default Khan Academy avatar avatar for user
    • aqualine ultimate style avatar for user Johnathan
      This is slightly tricky since you have a term with an xy. So here's what I'd do. I'd factor out the 5x's and leave the y's.
      5x(5 - 1 - 3y) - 3y
      5x(4 - 3y) - 3y
      If it was just -15xy-3y alone, I would just leave the x since the second term doesn't have an x in it.
      -3y(5x - 1)
      Like that.
      (5 votes)
  • leafers ultimate style avatar for user Avi Sanders
    Isn't the title of this video potentially misleading? The title is "Factoring two-variable quadratics: grouping", but the polynomial in the video 5rs+25r-3s-15 has no second degree exponent and is therefore a linear polynomial rather than a quadratic.
    (5 votes)
    Default Khan Academy avatar avatar for user
  • piceratops ultimate style avatar for user Andrew Hung
    I'm curious about how useful this type of technique is. It seems that the problems it can solve are very artificial - they conveniently happen to have a common factor (in this case, s + 5) that can easily be factored out. How often, when solving real-world problems that weren't designed specifically to illustrate this technique, are you going to be fortunate enough to have something where this technique works out so nicely? I feel like you'd almost always just use the quadratic formula. Can't think of any situation outside of homework where this video's technique would be useful, but if anyone can, please enlighten me.
    (2 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Parth Nallaparaju
    how to solve this? x^2-y^2/xy - xy-y^2/xy-x^2
    (1 vote)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user seani
    How does this apply to the real world?
    (1 vote)
    Default Khan Academy avatar avatar for user
  • mr pants teal style avatar for user brian.garcia.kgc
    what does foil mean?
    (0 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user taniaan1901
    what is the answer with the different factors ? b^2-2b+1-y^2
    (1 vote)
    Default Khan Academy avatar avatar for user
    • aqualine ultimate style avatar for user Dominik.Ehlert
      This equation can't be factored that easily, because you would have to add some components first. By adding components I mean taking something like + a - a to the equation. As a - a = 0, this doesn't change the value of the equation. In this case I add - b*y + b*y - y + y, so I can factorize the equation:
      b^2 - 2*b + 1 - y^2
      = b^2 - b - b + 1 - y^2 - b*y + b*y - y + y
      = b(b - y - 1) + y(b - y - 1) - 1 (b - y - 1)
      = (b + y - 1)(b - y - 1)

      You can check this is the solution by multiplying the factors again.
      (2 votes)

Video transcript

We're asked to factor this expression by grouping. Now, they mention grouping, we're going to see what grouping is, but we're going to see very quickly that we have to do this thing called grouping because you can't just factor this expression. If you look at these, each of the terms, all but one of them is divisible by 5. So you can't just factor out a 5. Not all of them are divisible by either r or s. This is only divisible by r, that's only divisible by s, that's divisible by neither. So there is no common factor across all four of these terms. That's why we have to group them into groups where there are common factors, and then see if that simplifies the whole thing. And there is a little bit of an art to recognizing when you can factor by grouping, but they've set this problem up nicely for us. So if you look at these first two terms right here. You have a 5rs and a 25r. These two guys clearly have some common factors. They're both divisible by 5, they're both divisible by r. So if I just wanted to factor this one out, or if I wanted to rewrite it as a product of two expressions, how could I write it? Well, I could write it as a product of 5r times-- what's 5rs divided by 5r? Well, you still have an s left over, you just have an s there. Plus-- what's 25r divided by 5r? Well, 25 divided by 5 is 5, and r divided by r is just 1. So 25r over 5r is 5. So these first two terms can be factored into these two expressions. And then let's look at these second two terms. Well, they definitely have a common factor, you have a negative 3 or positive 3 common to both of these. Let's just go with the negative 3. And our goal is really to factor it into a negative 3 times, hopefully, something very similar to s plus 5. And you might already be seeing that it's going to factor into s plus it. 5. So let's factor out that negative 3. So these two terms you can rewrite as negative 3 times-- what's negative 3s divided by negative 3? Well, you're just going to have an s left over. And then what's negative 15 divided by negative 3? Well, that's just positive 5. And just like that, we've grouped them and we're able to factor each of those groups, and then something interesting might pop out at you. And one, you can always verify that you factored this properly by distributing each of these expressions. Distributing the 5r times s plus 5, and the negative 3 times s plus 5, you'll get exactly this. But something maybe jumped at out you just now. You have 5r times s plus 5. Then you have negative 3 times s plus 5. So now this expression, we have two terms instead of four, right, this is one term, this is another term. And they both have s plus 5 as a common factor, so we can now factor out s plus 5. So this whole thing can be rewritten as s plus 5 times 5r. Right? If you take 5r times s plus 5, and you factor out the s plus 5, you're just left with the 5r. And then similarly, if you take negative 3 times s plus 5, and you factor out the s plus 5, or divide by s plus 5, you just have a negative 3, just like that. And then we're done! We've factored this expression by grouping. It's s plus 5 times 5r minus 3. And you can verify it by multiplying it out. If you distribute the s plus 5 onto each of these terms, you'll get this expression up here, and then if you distribute the 5r over there you're going to get that expression. If you distribute the negative 3, you're going to get that expression. So this does simplify to that, so we have factored