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Factoring quadratics: common factor + grouping

Sal factors 35k^2+100k-15 as 5(k+3)(7k-1). Created by Sal Khan and Monterey Institute for Technology and Education.

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  • leaf blue style avatar for user David966★_
    Are polynomials the same as quadracts?
    (8 votes)
    Default Khan Academy avatar avatar for user
    • mr pink red style avatar for user Mike Sutjipto
      Quadratics are a special kind of polynomial. Here are some examples of various kinds of polynomials:

      (1) x^2 + 3x + 9
      (2) x^3 + x^2 - 9x
      (3) x^5 - 5x^3 - 2x^2 + x - 20
      (4) x^10 + x - 1

      While each of the above is a polynomial, only (1) is called a quadratic -- this is because its largest exponent is a 2. Another way of saying this is that (1) is a "second-degree polynomial". (2) is a "third-degree polynomial" because its largest exponent is 3 (these kinds of polynomials are often called "cubics"). (3) is a "fifth-degree polynomial", (4) would be a "tenth-degree polynomial", and so on.

      So basically, the term "quadratic" is just another word for any second-degree polynomial -- or any polynomial where the largest exponent is 2.

      (Also, I used x as my variable in all the above examples, but you could use any variable you like... it doesn't need to be x)
      (50 votes)
  • blobby green style avatar for user scotty  peirce
    Okay, so we found out that this factors into 5(7k-1)(k+3), but what does k equal?
    How do you find out what k is when there's a 5 outside the factored terms?
    (10 votes)
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    • hopper cool style avatar for user Chuck Towle
      Scotty,
      The original problem was
      Factor 35k² + 100k - 15
      This is an expression, but it is not an equation. So you cannot solve the equation for k. All you can do is factor the expression.

      If instead the original problem was
      35k² + 100k - 15 = 0 you could slove for k
      You would first factor the left side, just like in the video and you would get
      5(7k-1)(k+3) = 0
      The expression on the right would be equal to zero when either (7k-1) = 0 or if (k+3) = 0 because anythng times zero is zero.
      And the solutions for k would be k=-3 and k= 1/7

      But that was not the solution to the expression in the video because it was not an equaltiy, all we could do is factor the expression.

      Factoring an expression may not seem like it has a purpose until you can then use it to solve something, but learning how to factor expressions is the first step to solving quadratic equations.

      I hope that helps you make some sense of things.
      (41 votes)
  • aqualine tree style avatar for user Gert Olivier
    Hi at - , Sal mentions finding a number whose product is 7 * -3. Why isn't he simply referring to the number -3 as his product?

    Thanks in advance.
    (11 votes)
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  • blobby green style avatar for user Irene Wu
    is this the same as factorising using common prime factors?
    (7 votes)
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  • aqualine ultimate style avatar for user Calvin Haworth
    I understand the concept, but I have a problem that doesn't make sense. If the highest exponent variable doesn't have a constant before it, how would I find the common factor?
    The problem is:
    y^4 + 10y^2 + 24=___

    Thanks in advance
    (4 votes)
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  • blobby green style avatar for user Nick Fealy
    Would using brackets be acceptable?
    (4 votes)
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  • leaf blue style avatar for user ASforbooks
    Are there by any chance any videos explaining how to factor problems like:

    2x^3 - 16x^2 + 24x

    If so, I would appreciate if someone could provide me a link to where I can learn how to solve these types of problems, or teach me themselves.
    Thanks a bunches!
    (1 vote)
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    • piceratops ultimate style avatar for user Just Keith
      2x³ - 16x² + 24x
      Step 1: Factor out the GCF, in this case it is 2x
      2x[x² - 8x + 12]
      Step 2; Factor [x² - 8x + 12]
      (x-6)(x-2)
      Step 3: put it all together
      (2x)(x-6)(x-2)
      If you need to know how to factor cubic equations that don't have a GCF to factor out that makes it easy, let me know and I can walk you through that.
      (11 votes)
  • blobby green style avatar for user mrrucker66
    Consider the polynomial function
    Px=x4-3x3+ax2-6x+14,
    where a is an unknown real number. If x-2is a factor of this polynomial, what is the value of a?
    (3 votes)
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    • mr pink red style avatar for user andrewp18
      Interesting question.
      Well since x - 2 is a factor of P(x), we know that 2 is a root of P(x). So we must have P(2) = 0. This means:
      2⁴ - 3(2³) + 4a - 12 + 14 = 0
      Now we can just solve for "a":
      16 - 24 + 4a + 2 = 0
      8 - 12 + 2a + 1 = 0
      -3 + 2a = 0
      2a = 3
      a = 2/3
      Comment if you have any questions.
      (5 votes)
  • blobby green style avatar for user Riley3894
    why do the questions have upvote and downvote buttons like Reddit?
    (2 votes)
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    • starky sapling style avatar for user KC
      So people can upvote the good questions so that other people with the same question can just read the response. The answers also have upvotes and downvotes so people can upvote the best answer.
      (7 votes)
  • aqualine ultimate style avatar for user c h i c k e n s
    Does it matter, in this example, if the 21k went first or the -k would. When I practice, I have no idea which one I should put first and sometimes get it wrong because of that. I'm really confused, please help!
    (3 votes)
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    • female robot grace style avatar for user C C
      It can matter. It's not wrong to put the -1k first, but sometimes you'll get stuck trying to factor like that.

      You want to think about the idea of "grouping" - which term groups better with it's neighbor. So 21k would be better to group with 7 k^2 because 21 is divisible by 7.

      In this example, 21 is also divisible by 3, so you can group the other way:
      5(7k^2 -1k + 21k - 3)
      5[k(7k-1) + 3(7k-1)]
      5(7k-1)(k+3)
      Here you get the same answer, but it might not always factor as neatly both ways.

      [Hmm, looking at the examples from the videos again, it seems like it would factor either way. Does anyone have an example where it doesn't? Or know the rule for it?]
      (4 votes)

Video transcript

We're asked to factor 35k squared plus 100k, minus 15. And because we have a non-1 coefficient out here, the best thing to do is probably to factor this by grouping. But before we even do that, let's see if there's a common factor across all of these terms, and maybe we can get a 1 coefficient, out there. If we can't get a 1 coefficient, we'll at least have a lower coefficient here. And if we look at all of these numbers, they all look divisible by 5. In fact their greatest common factor is 5. So let's at least factor out a 5. So this is equal to 5 times-- 35k squared divided by 5 is 7k squared. 100k divided by 5 is 20k. And then negative 15 divided by 5 is negative 3. So we were able to factor out a 5, but we still don't have a 1 coefficient here, so we're still going to have to factor by grouping. But at least the numbers here are smaller so it'll be easier to think about it in terms of finding numbers whose product is equal to 7 times negative 3, and whose sum is equal to 20. So let's think about that. Let's figure out two numbers that if I were to add them, or even better if I were to take their product, I get 7 times negative 3, which is equal to negative 21. And if I were to take their sum, if I add those two numbers, it needs to be equal to 20. Now, once again, because their product is a negative number, that means they have to be of different signs, so when you add numbers of different signs, you could view it as you're taking the difference of the positive versions. So the difference between the positive versions of the number has to be 20. So the number that immediately jumps out is we're probably going to be dealing with 20 and 21, and 1 will be the negative, because we want to get to a positive 20. So let's think about it. So if we think of 20 and negative 1, their product is negative 21. Sorry. If we take 21 and negative 1, their product is negative 21. 21 times negative 1 is negative 21. and if you take their sum, 21 plus negative 1, that is equal to 20. So these two numbers right there fit the bill. Now, let's break up this 20k right here into a 21k and a negative 1k. So let's do that. So let's rewrite the whole thing. We have 5 times 7k squared, and I'm going to break this 20k into a-- let me do it in this color right here-- I'm going to break that 20k into a plus 21k, minus k. Or you could say minus 1k if you want. I'm using those two factors to break it up. And then we finally have the minus 3 right there. Now, the whole point of doing that is so that we can now factor each of the two groups. This could be our first group right here. And so what can we factor out of that group right there? Well, both of these are divisible by 7k, so we can write this as 7k times-- 7k squared divided by 7k, you're just going to have a k left over. And then plus 21k divided by 7k is just going to be a 3. So that factors into that. And then we can look at this group right here. They have a common factor. Well, we can factor out a negative 1 if we like, so this is equal to negative 1 times-- k divided by negative 1 is k. Negative 3 divided by negative 1 is positive 3. And, of course, we have this 5 sitting out there. Now, ignoring that 5 for a second, you see that both of these inside terms have k plus 3 as a factor. So we can factor that out. So let's ignore this 5 for a second. This inside part right here, the stuff that's inside the parentheses, we can factor k plus 3 out, and it becomes k plus 3, times k plus 3, times 7k minus 1. And if this seems a little bizarre to you, distribute the k plus 3 on to this. K plus 3 times 7k is that term, k plus 3 times negative 1 is that term. And, of course, the whole time you have that 5 sitting outside. You have that 5. We don't even have to put parentheses there. 5 times k plus 3, times 7k minus 1. And we factored it, we're done.