@@MikehMike01 Well, you need to know the mathematical definition of what it means to factor (to write something as a multiplication problem) as well as what it means to factor a trinomial (the factors of the first and last terms should add up to the middle one). Beyond this, it's about trying out various algorithms and figuring out which one best suits you, based on your understanding of it. There are multiple algorithms (more than in this video), and even though I know a bunch of algorithms, even I have my preferences. Math is meant to be explored.
Best math teacher ever!!! He does an outstanding job. He explains it so well. Calmly and simply. Many methods to choose from. Thorough and detailed. With a dash of humor.
I think "Lazy AC" is a bad name, because "lazy" makes you feel bad for using it, but if it has been proven to work for all cases, why not use it? We dont have to prove, for example, that the area of a triangle is bh/2 everytime we use it, we just use it because it has been proven to work
I always come back to this video when I forget how to factor trinomials and I swear that the slide and divide method is the only one that works for me!! Thank you once again 🥰
There is a rigorous variant of the lazy A.C... Write T = the trinomial. Then on the next line multiply T by A so in your example you would get 4 T = ( 4 x + ___ ) ( 4 x + ___ ) This works because you multiplied x by A twice, but only need to do so once. Then proceed as in the video and you will find the extra A will factor out in one of two ways. It might factor out of one term as in your example, or you may find that both terms factor and the two factors together multiply to A. I prefer this method as it is easy to do and ALSO easy to understand why it works. But stepping back from my favourite, as a teaching technique I very much like your approach of giving students choices in how to do this--I think in a typical class you will have different people liking each method. It is good teaching and ALSO letting them discover their own sense of mathematical style
Thank you so much. This probably the best teaching video I have ever viewed. It is completely on-topic, describes everything in full detail, and now I can actually understand factoring binomials. You just saved a student from failing a quiz.
I have only recently discovered your channel. I am a returning college student after taking a few years off. It has been 4 years since I have taken any kind of mathematics course and this video has been a life saver to me! I am struggling to understand the basics of earlier math courses and it has impeded my studies. Khan academy videos and SPECIFICALLY this video have really helped me. Thank you :)
I wasn't taught any of these methods. I was literally just told to find two expressions, resulting in me doing a lot of sloppy guesswork. Made things unnecessarily frustrating, especially when A was not equal to 1.
Yeah, that's literally how they also used to teach it in textbooks back then, but, as mentioned in my textbook, many people hated that method, so people came up the AC with Grouping method.
I ended up sticking with the AC method, but I just want to say that the way you teach, with kindness, made me smile through the video. Thank you for being who you are and for helping us out!
I wasn’t able to understand this in class no matter how much I focused, but your way of teaching helped me learn how to factor trinomials in just 5 minutes!! Thanks for this! just subscribed
You are my HERO! I have cried and cried trying to figure this out and I don't know what I would have done without this help. You may have saved me from dropping out. THANK YOU!!!
The best in my opinion is the standard "factoring by grouping". Simple, safe, and won't cause confusion later on (in case you forgot how to do the other methods). However, the other methods are also nice, and your videos are very educating and entertaining for math geeks like me. By the way, have you checked the email I've sent to you earlier regarding the integral of sqrt( 1 + 81 X^4 ) dx? Wolfram Alpha shows me the integration result, but it's kinda odd, and includes elliptic integrals and that sort of things which I don't understand.
This guy totally ROCKS! I'm trying to help my 8th grade son and I haven't dealt with this math in 40 years! This guy made it happen! Wish I had him in high school! LoL
I'm 32 and going back to college. Your 4th and final method resonated most with me. Being able to check the math right there worked wonders. The amount of potential solutions threw me for a loop and caused me much frustration- but your video has helped me solve this riddle. Thank you for the guidance and advice.
There was a shout out for a proof of the Lazy AC Method. I've got one here: ua-cam.com/video/n4AgEd0DIZg/v-deo.html Check it out and please let me know what you think!
U are amazing man u helped me for my exam amazing........ I was preparing for my exam from other UA-cam channel but I did not understand but u explained me ...... So thank you for that
The Lazy AC method and 'slide and divide' are actually two variations of the same technique. The each rely on the zero product property insofar as either one or both factors of a quadratic expression are presumed to equal zero. Hence, if you can factor a non-zero number out of a binomial that is presumably zero whatever remains is then still equal to zero. I used to like 'slide and divide' best; esp. after the first time I saw this video. In the meantime I discovered Po Shen Lo's method of solving quadratic equations. I now prefer Po Shen Lo's method to factoring, CTS, or the quadratic formula.
I learned the first method as Product-Sum Decomposition and I still teach it to my students. I usually start with the simplified Product-Sum Decomposition, where the a-value = 1. y=x^2 + 12x + 35 Find two numbers that multiply to 35, and add to 12. I get 5 and 7. So the factor is: y=(x+5)(x+7) Then I go on to when a1. If all else fails, use the Quadratic Solution. It will find an answer is one is available. It also has the discriminant of the quadratic in it, so it can even tell you the number of solutions before doing all the math. Also, it is the method that will give you complex factors if that is your number space.
I do the same too! I watch other people teaching to get new ideas. UA-cam is amazing that allows me to learn from other teachers. Thank you both for watching my video.
@@StoopVital No teacher knows everything AND we should be able to honestly admit that. Maybe she just wanted to give her students more OPTIONS to choose from. A great quality in a teacher in my opinion.
Just been going over this with my daughter and realised I didn't have a method or rather .... I use trial and error to sculpt the correct answer. Thanks so much. I enjoyed going back to school again.
I do the lazy ac, but in a way that you can use the equal sign. So with your example. =((4x-8)(4x+3))/4 =(4(x-2)(4x+3))/4 =(x-2)(4x+3) By dividing everything by 4(or whatever the term for a was) straight away it's always equal to the original expression. There will always be a factorisation to get rid off the denominator.
The quadratic formula is not at all useless. Though using it to factor requires ingenuity. You get the answers for y = 0, but then have to figure out the coefficients. This can be quicker, but usually isn't.
ZipplyZane speak for. yourself. The square root is always of a perfect square if the trinomial is factorisable so its very fast to calculate the roots. You just subtract the roots from x within each factor. Then multiply by denominators to get integer values in the factors Speaking for myself i can always do this faster than any of the methods shown. Of course different folk have different skills and you have discovered that another technique works better for you. Thats cool too - just please do remember that it wont be the case for everyone...
I've never ever studied the methods in the video. Method number 1 was mentioned in my high school books but I applied it only once and already in college. Even when we had a subject all about factoring polynomials in college, we used the good ol' quadratic formula
THANK YOU!!! I have learned the 4th method in high school, I have always used it and it's AMAZING. I thought it was known. Anyways, years passed and I kind of forgot how to do it, and I got shocked when I tried to look it up because I couldn't find it anywhere. Everyone explains the other methods, no one knowns about the tic tac toe method... (my teacher used to call it a different name). I was really devastated because I wanted to remember it but I couldn't find anyone who knows about it, until I finally found your video!! Thank you!! I recalled it immediately I just needed this trigger, you're awesome. The tic tac toe method has always been the best.
Thank you very much! I've watched alot of the organic science tutor, khan academy, and more, after watching this video I realized that the Tic-tac-toe method is the best for me because of its similty after 15 Minuites I've learned more then I did in around 2 hours of youtube videos and 2 weeks of math class doing polynomials you did it slowly and showed us exactly what to do usually on other videos they would say something then I wouldn't understand so I would have to learn the concept but your explained them.
Once you get good with numbers, I think the Tic-Tac-Toe (or a version of it - have seen a couple) is good because it's easier to understand. It's a form of trial and error that directly relates to the mathematical definition.
OMG! The best method I've seen so far when factorizing quadratic equation! Very well-explained indeed... Lazy AC is my best choice. Thanks much brother...
Dear Sir, I like your math so much. I remember you once explained the cross method in different regions such as Taiwan, Hong Kong, and the United States in Chinese. However, I can't find this video now. Could you please tell me where I can find this video? Thank you!
OMG this was a huge help thanks so much!!!!!!!!!! I love the tic tac toe one it's really helpful and anyone could easily understand it. I was literally crying looking for videos to understand this concept and you made that happen, thank you so much!
I just did it by utilizing the 3B1B easy quadratic formula, which works even when its not factorable with rational or real numbers, and there's no guess and check required, you should check it out
I enjoyed the Tic-tac-toe method. As someone who can factor in their brain quickly it takes out a lot of the writing steps and leads to an easy to interpret final result (but any teacher can see how you got it)
@@ttsookoo lets say you have the polinomial 2x^2 -10x + 12, so a=2, b=-10 and c = 12. First, you set it equal to 0, then you solve for x and you get that the answers are 2 and 3. And in the end, 2x^2 -10x +12 = 2 (x-3) (x-2). In general, ax^2 + bx + x = a (x - 1st root of the polinomial) (x - 2nd root of the polynomial)
I really liked the Lazy AC. Very well explained. I will definitely be searching for more of your videos. I was really lost but just this video explained it all. Thank you.
Better to understand why and how factoring works than to memorize a method or trick to get the right answer. The point of factoring is to determine what two numbers are multiplied together when all you are given is the finished product of multiplication. The reason we factor polynomials is that polynomials represent a number, but with variables inside the number itself. The question becomes, how do we multiply numbers when we don't know what the variables inside a number are? The answer is that we distribute, so that even if the variable takes on any value, it does not change the result of multiplying by the number we distribute. Factoring asks the question: "what did we distribute by?" The answer is the GCF of the finished product of multiplication. When we factor, we are dividing our number by our GCF in order to reverse the process of multiplication. Then we write any number, including polynomials, in factored form by writing GCF * remaining values. The annoying part of this is that you have to know your multiplication facts well enough to be able to see what multiplied what in your head. This is what we do when we split a trinomial into a 4 term polynomial so that we can factor it with grouping. And by the way, factoring with grouping is precisely splitting up a polynomial with no uniform GCF into two groups that each have their own GCF's, then we factor each separate group in order to make an expression with does have one GCF for all terms, then we finish by factoring this last term to get the final factored product. This is exactly how and why factoring works. Now these tricks in this video can help make it easier for you to choose the correct two middle terms, but it doesn't make sense to learn the tricks before you understand what I just wrote above.
Robert Wilson III If I understand you correctly, what you are saying is something like this: If you start with a quadratic factorised into two linear terms, and then multiply out using distribution, we initially get 4 terms. Then when we combine the two terms in x to get a single x term and end up with a standard three term quadratic, we "hide" the original distribution process. So when we want to factorise we have to separate the x term back into two x terms to "reveal" how the distribution was done and so reconstruct the original factorisation.
I vaguely remember seeing the diagram, and of course the quad formular, but only use it for non trivial factors. I guess I use the 4th method, but in my head rather than formally. Yes, there is a choice of +/- 1 x -/+ 6 or +/- 2 x -/+ 3, but which one x4 plus the other results in 5? Can only be 4x2 and 3. Then determine the signs to get -5.
Alexis Caraballo That's exactly what you're doing when you use the "slide & divide" method. The quadratic formula always works, but you shouldn't need to use it when the original expression can be factored. That's kind of the point. Using the quadratic formula on a factorable polynomial is like using a jackhammer to crack a walnut. It's overkill.
This formula mtd is guaranteed to work. so under stressful conditions (in eg timed exams) when your brain just stops working, it might be more useful to rely on the quadratic formula mtd than guess n check. And I guess this mtd would be super useful when dealing with quadratic equations with complex roots
All these methods are just one and the same method presented in four different disguises :q The first method is just a fancy way to use Viète's formula ;) Still it requires some guess work, which is bad. Especially that it won't always work (e.g. when the roots are not integers it might be quite hard to factor) :P Also, instead of just guessing the numbers, it's better to just test the divisors of the `a·c`. Similarly the second method, which just uses the fact that the common factor of 4 that can be taken out from the second parenthesis is never 0, so it is not part of the solution. It's kinda similar to factoring out the 4 from the original trinomial equated with 0, and then, since the coefficients of `x` have a common measure (4), it works the same as with integers (the same solutions, just scaled up/down). Third method, again, is just Viète's formulae with scaling up/down by 4, as in method 2. Also the fourth method can be traced back to Viète's formulas, since it has the multiplications of `a·c` and summing up to `b` in it. You know what's the _best_ way of factoring polynomials? The one in which you don't have to GUESS the factors by trial and error! :P
Sci Twi yes all of these are the same in their underlying theory. as a fact of teaching rather than a fact of maths experience shows different learners find the different visual layouts easier or harder. Good maths teachers (including BPRP) offer muliple techniques so that students can choose the one that works for them. Bad maths teachers offer only one technique and leave half their students who that technique doesnt suit believing they are too stupid. Personally none of these work for me and I use the QF instead. But when I am teaching I offer a variety of methods and have observed that only a small minority of learners like my own favourite method. BPRP of course knows these four are all rooted in the same underlying algebra - but he does not let that get in the way of his teaching skills. The student does not need to know that the technique he likes is "just the same" as the one he hates. He or she does need to be encouraged to trust his own choices among a selection of valid techniques.
I just like to call the quadratic formula the ugly completing the square method. It's all the same, but really fun to see how many cool ways you can present it to make it look better for the ones that are learning.
4 methods, no one convinces me, the last one was the worth (guess, guess guess...) no, the only one that always works : is to find the roots of polynomial and (x-root1)(x-root2) roots can be fond by computing : delta=b²-4ac roots= -b +/- sqr(delta) / 2a that's done. no guessig at all.
You see after practising for a while, you can recognize when the solution is nice and clean (integers) or not. These methods are intended for fast and easy solving compared to using the quadratic formula. One personal favorite is to first compute that delta to see if it has real solutions. If it does, then try out the tic-tac-toe as shown in this video with a first guess (shit works most of the times) and if it doesn't, then completing the square and then a difference of squares (skipping some trivial calculations of course). It works just as fine as the formula, and unless we have very big coefficients (which I would just undeniably use the formula) it's faster. again, we only want to solve these problems as fast as possible.
Mine is similar to the 2nd method. However I used fractions. So we have the 2 factors of -24 which is -8 and 3 that adds up to -5. Now we divide the factors by a which is 4. So now we have -8/4 and 3/4. Now we simplify both fractions if possible. If the fraction becomes a whole number, put a 1 in the denominator. So now we have -2/1 and 3/4. The denominators are for the coefficients that are linked to the variable x and the numerator becomes the constant. So we will have (1x-2)(4x+3) or (x-2)(4x+3)
After wasting my time with several videos I found the one I was looking for. Thank you so much, keep up the good work. I finally got the concept of factoring.
Hello rpbp, I would like to know if the lazy ac method and the slide and divide method works if a is negative. I love your videos and your way of teaching. That has helped me a lot btw.
In principle, it should. In practice, if I need to factor a trinomial where a is negative, I find it easier to factor out the negative and go from there: -2x^2 + 5x - 2 becomes -(2x^2 - 5x + 2). If the factoring is for the purpose of solving a quadratic equation, it's even better: If -3x^2 + 8x - 2 = 0, then 3x^2 - 8x + 2 = 0 as well.
I can't tell you how many instructional math videos I've come across where it sounds like the tutor was recording through a tin can and string. Thank you for investing in a decent quality mic, it makes it a lot easier to focus on the content rather than just whats being said.
Nice! (x4) Also nice to find this now-5-yo gem!! My opinion is that the best method to use will depend on the coefficients of the given quadratic. And of course, all of these methods are practical only when the coefficients are integers. So I'll assume that's the case. I always like to start with trial-and-error on the simplest candidate zeros: Take a-b+c and a+b+c, to check whether ±1 are zeros. Then just check the discriminant, ∆ = b² - 4ac, to see whether it's a square. If it isn't, the zeros are irrational, so just use the quadratic formula. If it is, try one of your 4 methods. Fred
Learning these little devils now, and my teacher is teaching the grouping method and I am struggling I have to admit. My favorite is the Tic Tac Toe Method and I am going to show this to my class because I know I am not the only one struggling. Brilliant Mate! Thanks :)
Thank you a lot for this. Some of these were not taught when I was in school for some reason. Are there proofs of why these work? Especially slide and divide.
S&D (Slide & Divide) takes p(x) = ax² + bx + c and turns it into q(x) = x² + bx + ac. Suppose p could be factored into: p(x) = (dx + e)(fx + g) . . [where d, e, f, g are not yet known] Then a = df, b = ef+dg, c = eg You can check, by expanding the following product [from the first line], that: q(x) = (x + ef)(x + dg) So when you S&D, turning p into q, you factor q to get q(x) = (x + h)(x + j) . . [where h = ef, j = dg] by finding h & j to make that product agree [remember, at this point, we still don't know d, e, f, g] from which you can get the factorization of p as: (ax + h)(ax + j)/a = (dfx + ef)(dfx + dg)/(df) = [(dfx + ef)/f] (dfx + dg)/d = (dx + e)(fx + g) because all we need to know to do this, are a, h, and j, which we have. These give us the LHS, from which factoring a into d times f in different ways, will lead to values that will divide h and j, respectively. Fred
Great, because I did not understand the slide and divide method that our daughter is being taught. After seeing the AC method, I understand the magic. I think I like AC method because you can check each step ("is it equal? yes or no."). the other methods involve memorization without understanding, so I don't like them as much. Maybe o.k. for better students than me who can keep in mind the theory behind the other methods, but for me I need to see something where every step makes sense. Thanks Blackpenredpen for being so clear.
![Lp. Сердце Вселенной #30 ДВА В ОДНОМ [Один человек]• Майнкрафт](http://i.ytimg.com/vi/nrXy15orKhY/mqdefault.jpg)
I'm honestly surprised that this, being the first video that I clicked, was able to teach me how to factor better than my teacher. Thank you so much.
I am glad to hear!!!
@@blackpenredpen hoping it works for me too bc i am BEHIND rn
this video has the benefit of you already knowing everything your teacher taught you
@@MikehMike01 Well, you need to know the mathematical definition of what it means to factor (to write something as a multiplication problem) as well as what it means to factor a trinomial (the factors of the first and last terms should add up to the middle one).
Beyond this, it's about trying out various algorithms and figuring out which one best suits you, based on your understanding of it. There are multiple algorithms (more than in this video), and even though I know a bunch of algorithms, even I have my preferences. Math is meant to be explored.
@@chocolateangel8743 bruh too many words
11:11 "it's a fraction, that's no good"
oh man if I had a nickel for every time I had that thought on a test
Now what happens when you start having fractions of a nickel?
a nickel is a fraction of a dollar, so you my friend, just fricked yourself
slide and divide is the most horrible technique i've ever seen. doesn't follow math rules like lazy ac. but somehow it's even more work???
@@Tsunami0950I have a penny, 1/5th of a nickel.
You're a life saver dude. Thank you so much for this video! If I had you as a teacher I wouldn't mind math.
Best math teacher ever!!! He does an outstanding job. He explains it so well. Calmly and simply. Many methods to choose from. Thorough and detailed. With a dash of humor.
Excellent job. And of course the Lazy AC is my choice.
Rod Galloway thank you!
Finally found the first mathematician with swag
THANKSSSS!
Helped, thank you
Check out Cédric Villani, he got plenty of it :d
LMAO
Wtf even!? You clearly haven't seen enough mathematicians maybe
Thank you for this fantastic video !
The lazy AC is the fastest and the most practical in my opinion.
thanks
Welcome
URon M ,
Ron M but you have to make sure that the trinomial can be factored
I think "Lazy AC" is a bad name, because "lazy" makes you feel bad for using it, but if it has been proven to work for all cases, why not use it? We dont have to prove, for example, that the area of a triangle is bh/2 everytime we use it, we just use it because it has been proven to work
username: blackpenredpen
in the video: *uses chalk*
Username: Am I a joke to you?
r/usernamedoesntcheckout
yes i know there is no such subreddit, but there is one called r/usernamechecksout
I mean, he does use a red and white chalk so it does kind of work
whitechalkredchalk
funny...
I always come back to this video when I forget how to factor trinomials and I swear that the slide and divide method is the only one that works for me!! Thank you once again 🥰
What if the quadratic is normalized (i.e. a = 1) already?
I'm trying to apply the first method to 3x^2-11x-4, but after Foil-ing the final factors, it gets the wrong answer. Why is this?
@@muadhnate thank you! This was very helpful, I messed up after getting the mutiples of 1 and 12.
@@muadhnate so, I have a question. You go from 3x^2-12x+1x-4 to 3x(x-4)+1(x-4). Where did you get that -4 from in 3x(x-4) ?
There is a rigorous variant of the lazy A.C...
Write T = the trinomial.
Then on the next line multiply T by A so in your example you would get
4 T = ( 4 x + ___ ) ( 4 x + ___ )
This works because you multiplied x by A twice, but only need to do so once.
Then proceed as in the video and you will find the extra A will factor out in one of two ways. It might factor out of one term as in your example, or you may find that both terms factor and the two factors together multiply to A.
I prefer this method as it is easy to do and ALSO easy to understand why it works.
But stepping back from my favourite, as a teaching technique I very much like your approach of giving students choices in how to do this--I think in a typical class you will have different people liking each method.
It is good teaching and ALSO letting them discover their own sense of mathematical style
Thank you so much. This probably the best teaching video I have ever viewed. It is completely on-topic, describes everything in full detail, and now I can actually understand factoring binomials. You just saved a student from failing a quiz.
I like AC lazy the most.
I have only recently discovered your channel. I am a returning college student after taking a few years off. It has been 4 years since I have taken any kind of mathematics course and this video has been a life saver to me! I am struggling to understand the basics of earlier math courses and it has impeded my studies. Khan academy videos and SPECIFICALLY this video have really helped me. Thank you :)
I wasn't taught any of these methods. I was literally just told to find two expressions, resulting in me doing a lot of sloppy guesswork. Made things unnecessarily frustrating, especially when A was not equal to 1.
frozenbacon same here all I was taught to do is to basically do trial and error so I was just stabbing in the dark untill I got the answer
Yeah, that's literally how they also used to teach it in textbooks back then, but, as mentioned in my textbook, many people hated that method, so people came up the AC with Grouping method.
Exactly
@MsWilson
@@shaunakkulkarni4679 I can agree to that as long as the coefficient, a, equals 1.
I ended up sticking with the AC method, but I just want to say that the way you teach, with kindness, made me smile through the video. Thank you for being who you are and for helping us out!
I wasn’t able to understand this in class no matter how much I focused, but your way of teaching helped me learn how to factor trinomials in just 5 minutes!! Thanks for this! just subscribed
You are my HERO! I have cried and cried trying to figure this out and I don't know what I would have done without this help. You may have saved me from dropping out. THANK YOU!!!
The best in my opinion is the standard "factoring by grouping". Simple, safe, and won't cause confusion later on (in case you forgot how to do the other methods). However, the other methods are also nice, and your videos are very educating and entertaining for math geeks like me. By the way, have you checked the email I've sent to you
earlier regarding the integral of sqrt( 1 + 81 X^4 ) dx? Wolfram Alpha shows me the integration result, but it's kinda odd, and includes elliptic integrals and that sort of things which I don't understand.
This guy totally ROCKS! I'm trying to help my 8th grade son and I haven't dealt with this math in 40 years! This guy made it happen! Wish I had him in high school! LoL
turn caption on and read 7:52
thx i needed that
I'm down
OH.MY.GOSH
You won a Nobel Prize!
7:54 lets look at porn ey? can we reduce that?
6:35 *_draws boobs_*
I'm 32 and going back to college. Your 4th and final method resonated most with me. Being able to check the math right there worked wonders. The amount of potential solutions threw me for a loop and caused me much frustration- but your video has helped me solve this riddle. Thank you for the guidance and advice.
Thank you. And for your questions
I solved
(X+4)(5x-2).
Overrated comment
There was a shout out for a proof of the Lazy AC Method. I've got one here: ua-cam.com/video/n4AgEd0DIZg/v-deo.html Check it out and please let me know what you think!
As a teacher I just teach complete the square and quadratic formula. All this guess and check is mind numbing.
U are amazing man u helped me for my exam amazing........ I was preparing for my exam from other UA-cam channel but I did not understand but u explained me ...... So thank you for that
The Lazy AC method and 'slide and divide' are actually two variations of the same technique. The each rely on the zero product property insofar as either one or both factors of a quadratic expression are presumed to equal zero. Hence, if you can factor a non-zero number out of a binomial that is presumably zero whatever remains is then still equal to zero.
I used to like 'slide and divide' best; esp. after the first time I saw this video. In the meantime I discovered Po Shen Lo's method of solving quadratic equations. I now prefer Po Shen Lo's method to factoring, CTS, or the quadratic formula.
Thanks!
I learned the first method as Product-Sum Decomposition and I still teach it to my students. I usually start with the simplified Product-Sum Decomposition, where the a-value = 1.
y=x^2 + 12x + 35
Find two numbers that multiply to 35, and add to 12. I get 5 and 7. So the factor is:
y=(x+5)(x+7)
Then I go on to when a1.
If all else fails, use the Quadratic Solution. It will find an answer is one is available. It also has the discriminant of the quadratic in it, so it can even tell you the number of solutions before doing all the math. Also, it is the method that will give you complex factors if that is your number space.
That is great. Cannot wait to teach this to my middle school students.
you have to look up videos on something you were hired to already know?
I do the same too! I watch other people teaching to get new ideas. UA-cam is amazing that allows me to learn from other teachers. Thank you both for watching my video.
StoopVital LMAO
StoopVital They look up videos because maybe they aren't good at explaining it according to previous students.
@@StoopVital No teacher knows everything AND we should be able to honestly admit that. Maybe she just wanted to give her students more OPTIONS to choose from. A great quality in a teacher in my opinion.
Just been going over this with my daughter and realised I didn't have a method or rather .... I use trial and error to sculpt the correct answer. Thanks so much. I enjoyed going back to school again.
Three weeks of classes couldn’t teach me what he did in 5 minutes.
Ailith S it really is a shame
I do the lazy ac, but in a way that you can use the equal sign. So with your example.
=((4x-8)(4x+3))/4
=(4(x-2)(4x+3))/4
=(x-2)(4x+3)
By dividing everything by 4(or whatever the term for a was) straight away it's always equal to the original expression. There will always be a factorisation to get rid off the denominator.
My prefered method is 5: quadratic formula.
The quadratic formula is not at all useless. Though using it to factor requires ingenuity. You get the answers for y = 0, but then have to figure out the coefficients. This can be quicker, but usually isn't.
ZipplyZane speak for. yourself.
The square root is always of a perfect square if the trinomial is factorisable so its very fast to calculate the roots. You just subtract the roots from x within each factor. Then multiply by denominators to get integer values in the factors
Speaking for myself i can always do this faster than any of the methods shown.
Of course different folk have different skills and you have discovered that another technique works better for you. Thats cool too - just please do remember that it wont be the case for everyone...
me too
I've never ever studied the methods in the video. Method number 1 was mentioned in my high school books but I applied it only once and already in college. Even when we had a subject all about factoring polynomials in college, we used the good ol' quadratic formula
I'd prefer lazy ac to having to do a lengthy formula.
THANK YOU!!! I have learned the 4th method in high school, I have always used it and it's AMAZING. I thought it was known. Anyways, years passed and I kind of forgot how to do it, and I got shocked when I tried to look it up because I couldn't find it anywhere. Everyone explains the other methods, no one knowns about the tic tac toe method... (my teacher used to call it a different name). I was really devastated because I wanted to remember it but I couldn't find anyone who knows about it, until I finally found your video!! Thank you!! I recalled it immediately I just needed this trigger, you're awesome. The tic tac toe method has always been the best.
I am very happy to receive your comment Sara! Thank you as well!
This was much more helpful than my teacher's explanation. Thank you so much for your help!
Thank you very much! I've watched alot of the organic science tutor, khan academy, and more, after watching this video I realized that the Tic-tac-toe method is the best for me because of its similty after 15 Minuites I've learned more then I did in around 2 hours of youtube videos and 2 weeks of math class doing polynomials you did it slowly and showed us exactly what to do usually on other videos they would say something then I wouldn't understand so I would have to learn the concept but your explained them.
Once you get good with numbers, I think the Tic-Tac-Toe (or a version of it - have seen a couple) is good because it's easier to understand. It's a form of trial and error that directly relates to the mathematical definition.
I finnally found the BEST method in the world man the tic tac toe is so nice it works always and it is so easy to see your mistakes.
Tanks man!
8Med9Zonder I am very glad to hear it! : )
Yessss man the tic tac toe one
works if your fast at math
I am a newly hired professor. I teach remedial math. I will definitely show these methods to my students. Thank you. 😊😊😊
I'm honestly surprised that he taught this so much easier, simpler, and quickly, than my teacher could have.
OMG! The best method I've seen so far when factorizing quadratic equation! Very well-explained indeed... Lazy AC is my best choice. Thanks much brother...
It's actually the Australian method for factoring trinomials.
The first method was the best😀
Dear Sir, I like your math so much. I remember you once explained the cross method in different regions such as Taiwan, Hong Kong, and the United States in Chinese. However, I can't find this video now. Could you please tell me where I can find this video? Thank you!
OMG this was a huge help thanks so much!!!!!!!!!! I love the tic tac toe one it's really helpful and anyone could easily understand it. I was literally crying looking for videos to understand this concept and you made that happen, thank you so much!
Roziya Alla Noor I am very to hear. Best of luck in your class. : )
Thank you!
I totally understood this. I am doing online college and this was the best learning video I have seen. Very easy to understand. Thank you so much.
I just did it by utilizing the 3B1B easy quadratic formula, which works even when its not factorable with rational or real numbers, and there's no guess and check required, you should check it out
I was there yesterday
Lazy AC is my go-to from now on.
Thank you so much for this video!!!
for the ac method how come sometimes you still need to break it apart but sometimes thats the answer?
You saved my life!!! Online school and algebra factoring drills don’t go together 😭😭
My prefered method is the regular AC
Slide and divide is my favorite. It's just so much simpler for me. Thank you!!
Thanks so much! I liked the slide and divide one the best. The last one just seems pretty complicated with how they have to match up.
You're welcome! Thanks for watching as well = )
I enjoyed the Tic-tac-toe method. As someone who can factor in their brain quickly it takes out a lot of the writing steps and leads to an easy to interpret final result (but any teacher can see how you got it)
I was taught a different one: solve for X and then the polinomial is equal to a · (x-answer one) · (x-answer two)
Yeah but this works in every case.. you factor out in order to solve for roots really fast, solving for root to factor is pretty meh
Demonstrate an example please. Lol😅
@@ttsookoo lets say you have the polinomial 2x^2 -10x + 12, so a=2, b=-10 and c = 12. First, you set it equal to 0, then you solve for x and you get that the answers are 2 and 3. And in the end, 2x^2 -10x +12 = 2 (x-3) (x-2). In general, ax^2 + bx + x = a (x - 1st root of the polinomial) (x - 2nd root of the polynomial)
What do you do when the leading coefficient is one and can't be factored out?
@@chicchi1682 then a=1, so you still get p(x) = 1 • (x - 1st solution) • (x - 2nd solution)
Thank you , Mr.blackpenredpen my favourate slide and divide
I like the slide and divide :) Thank you :)
I really liked the Lazy AC. Very well explained. I will definitely be searching for more of your videos. I was really lost but just this video explained it all. Thank you.
Am I the only person thinking that he looks like he’s holding an ood translation sphere?
Bruh it's called a snowball mic
I love how this is the only tutorial on how to do something in math and is understandable
In less than a minute you explained what I've been struggling with for days. why does something so easy have to be so difficult?
Thanks a lot , you helped me more in 6 minutes than 2 teachers in the periods ( 2 hours)
You saved my ass, thank you so much! The AC method helps me a lot!
i am literally mind blown for the lazy ac method!! it just makes sense and its super easy!! thank you!!!!
Better to understand why and how factoring works than to memorize a method or trick to get the right answer. The point of factoring is to determine what two numbers are multiplied together when all you are given is the finished product of multiplication. The reason we factor polynomials is that polynomials represent a number, but with variables inside the number itself. The question becomes, how do we multiply numbers when we don't know what the variables inside a number are? The answer is that we distribute, so that even if the variable takes on any value, it does not change the result of multiplying by the number we distribute.
Factoring asks the question: "what did we distribute by?" The answer is the GCF of the finished product of multiplication. When we factor, we are dividing our number by our GCF in order to reverse the process of multiplication. Then we write any number, including polynomials, in factored form by writing GCF * remaining values. The annoying part of this is that you have to know your multiplication facts well enough to be able to see what multiplied what in your head. This is what we do when we split a trinomial into a 4 term polynomial so that we can factor it with grouping. And by the way, factoring with grouping is precisely splitting up a polynomial with no uniform GCF into two groups that each have their own GCF's, then we factor each separate group in order to make an expression with does have one GCF for all terms, then we finish by factoring this last term to get the final factored product. This is exactly how and why factoring works. Now these tricks in this video can help make it easier for you to choose the correct two middle terms, but it doesn't make sense to learn the tricks before you understand what I just wrote above.
Robert Wilson III If I understand you correctly, what you are saying is something like this: If you start with a quadratic factorised into two linear terms, and then multiply out using distribution, we initially get 4 terms. Then when we combine the two terms in x to get a single x term and end up with a standard three term quadratic, we "hide" the original distribution process. So when we want to factorise we have to separate the x term back into two x terms to "reveal" how the distribution was done and so reconstruct the original factorisation.
I vaguely remember seeing the diagram, and of course the quad formular, but only use it for non trivial factors.
I guess I use the 4th method, but in my head rather than formally.
Yes, there is a choice of +/- 1 x -/+ 6 or +/- 2 x -/+ 3, but which one x4 plus the other results in 5? Can only be 4x2 and 3. Then determine the signs to get -5.
Thank you so much!!!!!!
Nothing more satisfying than the matching pair in the grouping method.
For me it's easier finding the roots with the quadratic formula and then:
a(x - root1)(x - root2)
Good video
Alexis Caraballo That's exactly what you're doing when you use the "slide & divide" method. The quadratic formula always works, but you shouldn't need to use it when the original expression can be factored. That's kind of the point. Using the quadratic formula on a factorable polynomial is like using a jackhammer to crack a walnut. It's overkill.
Alexis Caraballo same but then I realized it’s not practical to work efficiently, so now I’m here
Grt
This formula mtd is guaranteed to work. so under stressful conditions (in eg timed exams) when your brain just stops working, it might be more useful to rely on the quadratic formula mtd than guess n check.
And I guess this mtd would be super useful when dealing with quadratic equations with complex roots
But if can use calculators in exams to solve for the roots of a quadratic eqn, then this would be the fastest and most reliable mtd
I think my favorite method is the AC method and I've been using it since I watched this video. Thanks so much!
(5x-2)(x+4)
Is that the answer of the 4x^2+ 18x+15?
Salamander It is the answer to the problem at 16:30.
I like the lazy AC method. I will try it tomorrow along with the tic tax toe method. I’ve been struggling with polynomials. Thank you!
WTF
I can do it right now
Thanks man;)
Slide and Divide is for sure my personal favorite, thank you!
I want the lazy ac because I am lazy.
Best factoring video I've come across, thank you!
All these methods are just one and the same method presented in four different disguises :q
The first method is just a fancy way to use Viète's formula ;) Still it requires some guess work, which is bad. Especially that it won't always work (e.g. when the roots are not integers it might be quite hard to factor) :P Also, instead of just guessing the numbers, it's better to just test the divisors of the `a·c`.
Similarly the second method, which just uses the fact that the common factor of 4 that can be taken out from the second parenthesis is never 0, so it is not part of the solution. It's kinda similar to factoring out the 4 from the original trinomial equated with 0, and then, since the coefficients of `x` have a common measure (4), it works the same as with integers (the same solutions, just scaled up/down).
Third method, again, is just Viète's formulae with scaling up/down by 4, as in method 2.
Also the fourth method can be traced back to Viète's formulas, since it has the multiplications of `a·c` and summing up to `b` in it.
You know what's the _best_ way of factoring polynomials?
The one in which you don't have to GUESS the factors by trial and error! :P
Sci Twi I have no idea why you have no idea that I actually have that idea.
Sci Twi
yes all of these are the same in their underlying theory.
as a fact of teaching rather than a fact of maths experience shows different learners find the different visual layouts easier or harder.
Good maths teachers (including BPRP) offer muliple techniques so that students can choose the one that works for them.
Bad maths teachers offer only one technique and leave half their students who that technique doesnt suit believing they are too stupid.
Personally none of these work for me and I use the QF instead. But when I am teaching I offer a variety of methods and have observed that only a small minority of learners like my own favourite method.
BPRP of course knows these four are all rooted in the same underlying algebra - but he does not let that get in the way of his teaching skills. The student does not need to know that the technique he likes is "just the same" as the one he hates. He or she does need to be encouraged to trust his own choices among a selection of valid techniques.
I just like to call the quadratic formula the ugly completing the square method. It's all the same, but really fun to see how many cool ways you can present it to make it look better for the ones that are learning.
All I have to say is...you're a life saver and a wonderful teacher!
My Person choice would be the tic tac toe method
That's my choice too.
honestly really loving this channel
Better than Khanacademy :)
Kristal Kristal no.......
Yes!
totally
Kristal Kristal noooo
thank you so much for this video. slide and divide helped me understand how to factor even when my teacher couldnt teach me. you saved my life!
4 methods, no one convinces me, the last one was the worth (guess, guess guess...)
no, the only one that always works : is to find the roots of polynomial and (x-root1)(x-root2)
roots can be fond by computing : delta=b²-4ac
roots= -b +/- sqr(delta) / 2a
that's done.
no guessig at all.
You see after practising for a while, you can recognize when the solution is nice and clean (integers) or not.
These methods are intended for fast and easy solving compared to using the quadratic formula.
One personal favorite is to first compute that delta to see if it has real solutions. If it does, then try out the tic-tac-toe as shown in this video with a first guess (shit works most of the times) and if it doesn't, then completing the square and then a difference of squares (skipping some trivial calculations of course).
It works just as fine as the formula, and unless we have very big coefficients (which I would just undeniably use the formula) it's faster.
again, we only want to solve these problems as fast as possible.
Mine is similar to the 2nd method. However I used fractions. So we have the 2 factors of -24 which is -8 and 3 that adds up to -5. Now we divide the factors by a which is 4. So now we have -8/4 and 3/4. Now we simplify both fractions if possible. If the fraction becomes a whole number, put a 1 in the denominator. So now we have -2/1 and 3/4. The denominators are for the coefficients that are linked to the variable x and the numerator becomes the constant. So we will have (1x-2)(4x+3) or (x-2)(4x+3)
After wasting my time with several videos I found the one I was looking for. Thank you so much, keep up the good work. I finally got the concept of factoring.
I love you.
How come a half a week of my teacher teaching this isn't as good as watching 6 minutes of this.
Hello rpbp, I would like to know if the lazy ac method and the slide and divide method works if a is negative.
I love your videos and your way of teaching. That has helped me a lot btw.
In principle, it should. In practice, if I need to factor a trinomial where a is negative, I find it easier to factor out the negative and go from there: -2x^2 + 5x - 2 becomes -(2x^2 - 5x + 2). If the factoring is for the purpose of solving a quadratic equation, it's even better: If -3x^2 + 8x - 2 = 0, then 3x^2 - 8x + 2 = 0 as well.
I can't tell you how many instructional math videos I've come across where it sounds like the tutor was recording through a tin can and string. Thank you for investing in a decent quality mic, it makes it a lot easier to focus on the content rather than just whats being said.
I’ve watched tons but they didn’t really help, this was the most helpful, thanks a lot
Your a good teacher you inspired me
Nice! (x4) Also nice to find this now-5-yo gem!!
My opinion is that the best method to use will depend on the coefficients of the given quadratic.
And of course, all of these methods are practical only when the coefficients are integers. So I'll assume that's the case.
I always like to start with trial-and-error on the simplest candidate zeros: Take a-b+c and a+b+c, to check whether ±1 are zeros.
Then just check the discriminant, ∆ = b² - 4ac, to see whether it's a square.
If it isn't, the zeros are irrational, so just use the quadratic formula.
If it is, try one of your 4 methods.
Fred
Is there any downside in using the Lazy AC method vs the standard method?
Learning these little devils now, and my teacher is teaching the grouping method and I am struggling I have to admit. My favorite is the Tic Tac Toe Method and I am going to show this to my class because I know I am not the only one struggling. Brilliant Mate! Thanks :)
Like them all. Makes it not so boring having variety of options. Grid method though is my favorite. Makes you feel like your about to play a game.
Thank you a lot for this. Some of these were not taught when I was in school for some reason. Are there proofs of why these work? Especially slide and divide.
S&D (Slide & Divide) takes p(x) = ax² + bx + c and turns it into q(x) = x² + bx + ac.
Suppose p could be factored into:
p(x) = (dx + e)(fx + g) . . [where d, e, f, g are not yet known]
Then a = df, b = ef+dg, c = eg
You can check, by expanding the following product [from the first line], that:
q(x) = (x + ef)(x + dg)
So when you S&D, turning p into q, you factor q to get
q(x) = (x + h)(x + j) . . [where h = ef, j = dg]
by finding h & j to make that product agree [remember, at this point, we still don't know d, e, f, g]
from which you can get the factorization of p as:
(ax + h)(ax + j)/a = (dfx + ef)(dfx + dg)/(df) = [(dfx + ef)/f] (dfx + dg)/d = (dx + e)(fx + g)
because all we need to know to do this, are a, h, and j, which we have.
These give us the LHS, from which factoring a into d times f in different ways, will lead to values that will divide h and j, respectively.
Fred
It's basically just compensation.
Excellent explanation of each method. I prefer the first one.
The Lazy AC method is the fastest and easiest! The AC and grouping is second best! Thankyou! I'm studying for the GMAT
The lazy AC Method really helped me a lot.
Yessss! my teacher couldn't seem to teach the X method right, and that helped so much! Your a life saver :))
Great way to present ways to factor trinomials. Thanks.
Great, because I did not understand the slide and divide method that our daughter is being taught. After seeing the AC method, I understand the magic. I think I like AC method because you can check each step ("is it equal? yes or no."). the other methods involve memorization without understanding, so I don't like them as much. Maybe o.k. for better students than me who can keep in mind the theory behind the other methods, but for me I need to see something where every step makes sense. Thanks Blackpenredpen for being so clear.
I am very happy to help!! Thank you Alicia for your nice comment.
my new favorite method is slide and divide, thanks for the video!