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.

# Intro to grouping

Sal introduces the method of grouping, which is very useful in factoring quadratics whose leading coefficient is not 1. Created by Sal Khan and CK-12 Foundation.

## Want to join the conversation?

• What about a problem with four terms and no common factor where you have to factor by grouping, like: 3x^3 - x^2 +18x - 6? My book only gives examples with 3 terms and a common factor and the gives problems at the end of the chapter like that one...
• In this case you factor as he did after he went through his little process to create four terms, but you don't do that little process. You group the terms: (3x^3 - x^2) + (18x - 6) and factor out what you can from each term: x^2(3x - 1) + 6(3x - 1). Now you go on and factor out the common factor: (3x - 1)(x^2 + 6). I hope this answered your question, I was a little iffy on what exactly you meant.
• At , wouldn't ' fx * hx ' be equal to ' (f*h)*x^2 ' (f times h times x-squared) instead of just ' fhx ' ?
Take ' (2*3) * (2*4) ' , for example: that would be ' (3*4)*2^2 ' (3 times 4 times 2-squared), NOT ' (3*4)*2 ' , right?
• Indeed it should be fhx^2. There's an annotation pointing out the error a few seconds later, though.
• Why does he multiply 4 by -21?
• He is solving using this trick. The mystery numbers (a,b) must be multiplied to get -84 (4 times -21) and when added, will get 25 (the middle number).
• Does Sal mean "a x c" since the equation is ax^2+bx+c and does he mean simply "b" because 4+25= 29. I learned this in class but I wanted to make sure. So can someone clear this up you can find it in to
• What is the significance of learning this method of grouping? Why not learn how to factor these quadratics with the quadratic formula?
• When you practice this and other methods of grouping/factoring, your intuition of how to factor and what factors are likely will grow so that you won't need to use the quadratic formula, you will be able to "see" or "intuit" what the possible factors could be. Surprisingly, a high number of students make careless errors using the quadratic - it is best to use it only when all else fails.

Developing your mathematical intuition is the best thing you can do for yourself when learning math.
• Did Sal mistook at ? I think fx times hx is fhx^2 not fhx
• You are correct, but this mistake has been corrected with a text box in lower corner within 2 seconds of the error. It does not show up if you are on full screen.
• it is so satisfying to fully understand something that is a bit challenging
• fhx should be fhx^2
• Yes, the correction box at shows this (it does not show correction boxes if you are on full screen mode).
• The only thing I don't fully understand is how it works out that we multiply "c" or the constant in the polynomial by the coefficient of x^2. I didn't understand when Sal was explaining using all letters at the 9 minute mark. I tried reverse engineering by starting with a quadratic with an "ax" other than 1, but I still don't see how it means that the last term needs to be multiplied by the leading coefficient. The last term is not made up by multiplying it with any x term when expanding. Example: In 3x^2+17x+10. Why would the 10 need to be multiplied by 3? I expanded this from (3x+2)(x+5) and I still don't see it. Please help?
• (3𝑥 +2)(1𝑥 + 5) = (3 ∙ 1)𝑥² + (3 ∙ 5 + 2 ∙ 1)𝑥 + (2 ∙ 5)

As we multiply the coefficient of the 𝑥²-term with the constant term we get (3 ∙ 1) ∙ (2 ∙ 5),
which we can also write as (3 ∙ 5) ∙ (2 ∙ 1)

Now compare this to the coefficient of the 𝑥-term, 3 ∙ 5 + 2 ∙ 1

We have found that the product of the coefficient of the 𝑥²-term and the constant term is also the product of two other numbers whose sum is equal to the coefficient of the 𝑥-term.
• I've rewatched this video so many times. I still don't understand why this works or what I'm even supposed to do in order to factor out an expression like this. Why does ab need to be equal to -84? Why do we even multiply a and b together, or 4 and -21? I'm totally lost, could someone please point me in the right direction with a better resource/less confusing explanation?
• The point is that we need to split the middle term into two middle terms, with coefficients a and b, in such a way that 1) the four-term expression is equivalent to the trinomial, and 2) the four-term expression can be factored by grouping.

To satisfy 1), we clearly need to make a+b equal to the coefficient on the middle term of the trinomial. In this particular problem, we need a+b=25.

To satisfy 2), we need to have the ratio of the first term to the second term equal to the ratio of the third term to the fourth term (otherwise, there won't be a common factor to factor out in the final step of factoring by grouping!). This means that we need the product of the two middle coefficients (i.e. ab) equal to the leading coefficient times the constant term. So in this particular problem, we need ab=4(-21).

Have a blessed, wonderful day!