FASTEST way to factor a trinomial!
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- Опубліковано 18 гру 2020
- Original video: 4 popular ways to factor a trinomial: • 4 popular ways to fact...
This is called the "Slide-and-Divide Method" that I learned from my colleague, Prof. Tchertchian!
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The title looks so familiar 😂
Eyes, I copied it from some guy named Dr. Payment!
And I am still waiting for him to comment on this video so I can pin his comment. BUT I guess I will just pin yours instead.
Hahahahaha
😂😂😂
@@ViratKohli-jj3wj Virat Kohli 🥺🥺 aap bhi math krte hain Kya,
This really changed my life , I’ve never been so confused 💀
Different.
Lol 😂
Lol same
Jajajajaja *laughs in spanish
yh ikr haha
This tutorial needs a tutorial 💀
After doing some research, I figured out why it works the way it does. Firstly, you should know that the reason you're able to multiply AC is because of the nature/pattern of Quadratics. No matter the method you use, the factors you generate will always be factors of AC.
Secondly, this method works by compensation. When you multiply AC, you're actually increasing the constant by whatever the leading term is. This also means that the factors you generate are going to be as many times as large (based on whatever the leading coefficient is. As such, you have to divide by whatever the leading coefficient is to compensate for this.
Now, I want to address by you swing up the leading coefficient when it can't be divided out. What you're actually doing is multiply the fraction by whatever the denominator is -- in order to put it in standard form. Technically, leaving it as a fraction is not wrong. The fraction and standard form are actually equivalent. I hope this helps.
Not work in everywhere
Actually quite simple once you get it. And it's much, much faster then the normal method's, at least for me.
@@rocketpencil5948 It is simple, once a person "gets it." However, one problem I continually see is that too many math will call things "tricks," don't explain the mathematical logic behind what they are doing, and just expect students to memorize steps. The longer approach isn't much longer than the shortcut, and you can actually see your work. Using area models are pretty quick as well.
ts so simple bruh ☠️
Literally been tryna figure this out for an hour and a half and this dude came in with a min video and randomly all makes sense💀
This is why you need to focus and learn the fundamentals.
Real💀 I was already thinking about life after failing my math exam
THANK GOD HE MADE THIS!!!!
I have some steps, label each trinomial as eithrr a b or c (going in order) multiply a and c factor the resulting expression, devide everything by a. If a term cannot divide by a slide a infront of the variable. Hope this helps
Better way and makes more sense:
Do A times C because quadratics are in the form ax^2 + bx + c
So 4 x 6 = 24
Factors that make up 24 and sum to -5 is -8 and 3, So rewrite original equation as
4x^2 - 8x + 3x - 6
Then factorise
4x(x-2) + 3(x-2)
Factorise again
(4x-3)(x-2)
What. What is this monstrosity. Who figured this out. My brain has completely deflated due to utter confusion.
Hi
@@alvinbowley4230 Hi.
@@RJ-sm2cg Math teachers arent there to teach the easy way, they are supposed to teach you so that you understand *why* the math works, if all they ever did was give you the formula that you plug things into, it would be pointless and you would leave the class having learned nothing
@@RJ-sm2cg Teacher is supposed to teach you how and why math works. You can find formulas in textbooks or on the internet, so there is no reason to memorize them. Btw everything up to 12th grade level is easy enough that if you take your time to understand it deeply, you won't have to know any formulas
@@TabbyVee You say that but were taught so many methods when we’re younger that we have to forget bc it was just an easy way of teaching.
I LOVE THIS! Why was I not taught this? So many hours lost to needless trial-and-error factoring and quadratic formulating that I’ll never get back. This is BRILLIANT.
BRO SAME. This is like genius!
i learned it when i first started to factor, however it doesn’t always work i don’t think. But completing the square is my go to
The fact that this is so much easier
Same 🤫🧏
your feelings are irrational
You just explained what took my teacher 2 weeks to explain to my class.
you have all of that background information from the two weeks of class and this is one small, simple method of an entire family of concepts
bruh what my teacher explained it in 1 class and everyone understood
@@maxpie16 my teacher spent a whole month on matrices. im serious.
@@_quixotewhy though you can learn it in a few hours.
@@maxpie16it depends on your teacher, mine barely teaches, she just did one class and assumes everyone understood, we always get 6-13 over 20 as our score because of that.
I have an undergrad degree in mathematics. I feel cheated that I never learned this! All my future students that I tutor will be thanking you for sharing this!
this is the intuition for the AC method
But I thought most schools teach this
Really? They usually teach the AC method in Algebra 1
Can you explain how or why this works? Like, what's going on under the hood here, how would I ever know to do this?
@@astralbeatz9950 A self contradiction!
It's a perfect method I have been using it since your 100 Trinomials video came out. Thank you for it!!
Amazing! : )
I've been using it from 6th, it's called splitting the middle term. Yes Im an asian🤣
@@zerksez9963 stop saying i am an asian on every comment only the most stupid people brag about their knowledge
@@aayushswami3022 Is splitting something to brag about. I was just lettin them know how much better asians specially indian people are then them in calculating and yet we are not recognised so much. I don't think splitting is something to brag about. Even simple Calculus isn't, I don't know, maybe u should fix ur Preassumptions bout someone.
@@zerksez9963 bruh thats literally bragging be a little bit humble brother yes its a stereotype that indians are bad at everything but we dont need to prove it to anyone
another method:
P = 4 (the coefficient in 1st term)
Q = -6 (the coefficient is 3rd term)
P × Q = -24
Try to find 2 numbers when added is equal to the coefficient of the middle term (which is -5) and if multiplied then equal to P × Q (-24)
You can do this by the process of elimination or do this in a calculator which is easier. (Scroll down to see how)
In this case it is -8 and 3 which adds to -5
And -8 x 3 = -24
So. 4x² - 8x + 3x - 6
and then 4x(x-2) 3(x-2)
Last step (x-2)(4x+3)
Too smart for me
yes, the middle term breaking
Yes ! It is how I learned it
THANK YOU SO MUCH
Yeah dont understand and im a junior in highschool with a math final tomorrow
So glad I ran into my algebra teacher at an adult store, she passed me with a C- just to make sure I didn’t have to retake the class with her again.
At an adult store? 😭
I'd show her my thanks if i were you, if you know what i mean ;)
omg what
Lucky
Math really is hard 😂
Any *legal* proof for this?
Yes. (Hint : Quadratic Formula. Do you see that the discriminants are equal ?)
@@donaldbiden7927 he doesn't want just this concrete equation.
He wants to prove that the method works for every factorable quadratic equation.
He does it pretty similar to the way I know how.
Quadratic equation:
ax^2 + bx + c
Let 2 factors of a×c be p and q such that:
p × q = ac
p + q = b
ax^2 + px + qx + c
px(qx/c + 1) + c (qx/c + 1)
(qx/c + 1)(px + c)
(qx + c )(px + c)/c [qx/c + 1 = (qx + c)/c]
Using blackpenredpen method:
ax^2 + bx + c
write as :
x^2 + bx + ac
Let two factors of a×c such that:
p × q = a×c
p+q = b
Then:
x^2 + px + qx + ac
(x + p/a) (x + q/a)
Put a = pq/c
(x + cp/pq) (x + cq/pq)
(x + c/q) (x + c/p)
(qx + c ) (px + c) / pq
(qx + c) (px + c) / ac
ignoring a in denominator
(As bprp does in last step)
(qx + c) (px + c) / c
sloppy ending but is this good enough?
@@LetsSink you are the mvp
I love how he builds up hype and all in the end by saying "THIS IS IT!" and then just normally says "done"
Find 4 x -6 = -24
And then what two number you would add and get -5 and multiply them and get -24
-8 and 3
And boom
😎
SAVED MY LIFE ON THIS QUADRATIC QUESTION MAY THIS CHANNEL GO FAR.
That is much faster! That was awesome!
I'm going have to use this next time I have to factor. Thank you!
That's called Splitting the Middle Term/Middle Term Break,learned it in 8th Grade, Super useful
Omg same it's also called middle term break in my area I learnt it in 7 grade
@@rc-shadow9691 are u from asia?
@@abd_sh_321 ye
@@rc-shadow9691 which country
I am from pakistan,we learnt it in 8th grade
We were taught this method in our school in 6th grade. Yes I'm Asian 😂
Yep what's so interesting in that
Same here 😂
I came here looking for indian comments 😂😂
@Wayne Bhai yep man😂
@@anuragsinha3670 ah I see,a man of culture
Finally someone is using an easy method
I think I know why this makes sense. So basically, you take the quadratic equation formula, where it's and you can move it from a to c, and it doesn't change value under the square root, only decreases the denominator, so you can solve that by decreasing the values by the correct amount
Asslam o alikum. We need to find 2 values that multiply to -24 and add -5.
4x^²-8x+3x-6. So we use the grouping method.
4x(x-2)+3(x-2).
So the answer is:
(4x+3)(x-2).
I think he did something similar to the grouping method.
Have a good day.
… this is what i have always needed in my life
😮
Asian kids attendance here!!!!
We legit are taught this in school😁
This was a 6th grade indian question 😂
We got this in 6th grade in India
Great. But can that make India a great nation? I think not 😅
I'm not Asian but I learned this in school I think everyone did except for Americans
What school did you go to, bruh 😂😂
This confused and amazed me at the same time💀 i remember 2 years ago me and my friend were doing a similar thing, she obviously knew so much more than me and she was the one who gave the main idea of it. But our teacher said that it just doesn't make sense and even said that she would have to make us leave the class if we don't respect the rules💀 i don't know where my friend is right now, but that just reminded me of her, she was a very kind, smart and lovely person! I can now practice this method too and get reminded of her every single time lol! 💕
A good memories !
Oh yeah I also remember when I was small I was damn scared when I was instructed to sit between two girls.
But after sometime we started eating lunch together, talking together. and solving some sum together and now sometimes it reminds of those days
Those good days when we were just innocent kid
@@xninja2369 wow that was so cute and wholesome! 🥰
@@thebox7673 thanks bro
Your one was wholesome too,
Hope you would be doing great :)
thats a bad teacher. y'all shoulda not been threatened for doing things a different way just cuz teacher didn't understand it
@@OpLapDancePikachu69 agree you
I was taught this in school, the moment I saw the title I figured he was gonna talk about the slide and divide method
I legit felt my life changing 🙆♂️🔥
Bruh as an Asian why this is in my text book that I'm just reading now.
Am I the only one who learned this in school but had to watch over 5 times to understand that's what he was explaining?
Yep, I’ve been doing this since 8th grade, but I also didn’t understand the method he was using at first. But I’ve never actually written a new polynomial for it so.
It’s splitting the middle but more complicated
I am almost never given trinomials with a non-1 coefficient for the first term, but if I ever am, I am prepared.
You: Make me confused
Khan Academy: Literally puts the information inside my brain💀
I've been using this my whole life 😂
@@Animesh_Chaturvedi LoL f
are you a teen?
Same
Story of every indian
Does that means you do not know how to use the scientific calculator ?
I learned the ac method. You multiply a and c (a is the leading coefficient c is the constant) and find the numbers that add/subtract to the product. Then you put them into a factored group with the leading coefficient and divide by the leading coefficient and then simplify. For example: 3x^2 -4x -15; ac: -45 -9 + 5; 1/3(3x - 9)(3x + 5); (x-3)(3x + 5)
This is a fast way but you need to understand the other ways like factoring completing the square , etc.
This is because you multiply a and c in the -4ac, and then you need the division by a
So nostalgic. I remember doing a demo during Grade 7 of this method when I first studied in a western school. Damn I suddenly felt old.
This looks like a very unique interpretation of Vieta’s formulas.
Is it? Would be interesting to see how someone could prove this works
@@nikey2110 I probably wrote this while I was half asleep in the middle of the night, because this is definitely not Vieta's formula xD
Bravo 👏🏾 I honestly would’ve needed this in college 😂 it would’ve saved me a bunch of headaches lol
What an amazing way!
If I was taught this... it woulda been 30something years ago... but somehow I learned to factor differently. At this point, it would be good to be able to retain this newly learned method... but my brain is now cluttered with having to remember 400,000 online userids and passwords... so badly that I can't remember what I ate for breakfast.
Seems to me you need the help of a password manager.
@@Kettwiesel25 - Seems to me I need to embrace my inner Amishness 😆
now for a word from our sponsor
400,000 userids and passwords? what you do
400,000 userids and passwords? what you do
This dude literally just solved my decades of doubt in a minute
‘Here’s a super easy way…’ *quickly explains something super complicated* lmao
YOU ARE A LEGEND
Haha I independently discovered this method more than two years before this video, when I was teaching my son. Too many "degrees of freedom" when dealing with two coefficients you have to factor and juggle, so I wanted to make things simple for my kid by transforming it to a monic (lead coefficient one). I was inspired by the "ac" term in the discriminant of the quadratic formula, so thought of multiplying. And the rest of it is the same as yours. :)
well I discovered this method 3 years before this video so I am a better person than you
@@MrZootSuitz Well, bully for you. Your medal's in the mail.
I love his enthusiasm, there are also so many other ways to factorize! :))
Tysm I understood this the 1st time I watched it, and it saved me a lot of time for my super rushed math final (my teacher literally gave us 7 new topics to understand the day before).
The fact that this even works is honestly insane.
Omg that’s actually genius
Wait, is everyone not taught this? 😂
Middle term split??
Indians Rock 🤟🤟
Yeah 😄
@@chetanya_sharma yeah BOI it's taught in 8th class nowadays in India
@@aayushswami3022nope in 8th grade Shri dharcharya method is used to teach
I m indian too and the fact ppl here are considering it a different method shows that u guys rote memorize maths. This is not a new method it is basically saying that the constant is (a.b) and the middle term is -(a+b) in a general quadratic equation of the form (x-a)(x-b)=0 which is taught everywhere in the very first class of Quadratic equations. It is the simplest property of Quadratic equations. If u r calling it a new method and giving it an entirely different name like "middle term split method" that shows that u don't actually understand it in depth. For u it is a new trick for ur competitive exams. So come down off ur high horse.
I love your energy!!
The sound affects are what really helped me
My brain:error error error error error
I learned this in Algebra I at my school. We called it the A-C method, because you multiply the A and C terms of the quadratic.
Saved my life
Thank you, sir
My asian brain:
ToO eAsY
what u mean bruh i am asian and i just got 2.7 in math😢
This would have changed my life in school when I took maths instead of language.
CRAAAAZY BROOO I’ve always hated factoring, and even my schoolmates, we’re so bad at it the teacher even mentioned it lmao.. but this is crazy, i finally got it!
This was how I learned it at first. Completing the square made things so much more complex when I got to Pre-Calc, imo.
@nonchalant if you’re ever confused in calc, take the derivative of it. If that doesn’t work, integrate it instead. If that doesn’t work either, do the second derivative. If none of that worked, retrace your steps, you might’ve missed something.
Precalc… that was like 3-4 years ago, my memory isn’t that good, so I don’t have any good advice.
Always know he's Chinese from how he boxes stuff
He does the left side (shu), then the upper and right side (heng zhe), and then the lower side (heng)
That's some Sherlock level detection sh*t right there.
I love this method with my heart. Once I learned this in Algebra 2 I never used anything else lol
After seeing some of these people that have went through college not knowing this, I'm so glad I'm learning this now, in my first semester of college.
I’ve been using the box method for as long as I can remember, I’ll be sure to keep this method in mind. It actually works pretty well too, even on some complex problems. Why didn’t this show up in my recommendations earlier?
What? 💀 wth is that method? i’ve never been confused before tho-
This is called factorisation method.
This is not a trick it's a proper method.
Let's just take an example of a quadratic equation 3x^2+4x-4
In this equation the product of the coefficients of x^2 and constant is -12 so you have to make the factors of -12 suck that the sum of the factors will become the coefficient of x
Here it is 6 and -2
Now we can also write the equation as 3x^2+(6-2)x-4
=3x^2+6x-2x-4
Now you have to take 3x common in first two terms and -2 in next two terms
=3x(x+2)-2(x+2)
Now just take x+2 as common
You'll get (x+2)(3x-2)
And these are the factors of the polynomial
I literally accidentally found out about this method a few years prior while I was doing some questions in my math lesson. Thought I was a genius and discovered something only to be met by reality letting me know it's been around for a while.
I remember doing the same back in grade school when trying to calculate the square root of numbers. I got really close to the real answers. I don’t remember the method too well anymore, but I saw a video on YT that looked like what I did. No genius here :|
still sticking with my trusty old quadratic formula
When this method works, it is always faster than the quadratic formula. If you want to learn more about this, google "factor by grouping"
@@randomoneforstuff3696The key part 'when it works'
Thank you sir!!
How you get (X_8)and (X+3)
From usual factorising method
x²-5x-24 = x²+3x-8x-24
The rest should be obvious
With this knowledge I'll be able to take over the world!
The world of Trinomials...
Find 4 x -6 = -24
And then what two number you would add and get -5 and multiply them and get -24
-8 and 3
And boom
😎
My teacher called me stupid for not using his method and marked my problem solving wrong 😂😂
A straightforward way for those not getting it is to have the 4 be brought down when factoring already, then reduce by Greatest Common Factor.
(4x-8)(4x+3)- GCF of 4
(X-2)(4x+3)-solve for x
Thank you very much, now I will be able to power through the 100 or so textbook questions asking to factor non monic trinomials.
COOL! However, be sure to factor out any common factor first otherwise the common factor will dispear.
This helped me sooo much! I literally went from "I can't factor consistently" to "I can factor in seconds"
As someone who already used this method. I have never been so confused in my life
I love this! That said, if both of the factors have a coefficient on x other than 1, it’s a little more complicated.
Usually you look for the factors in the first coefficient and the last, then look for the combinations to see which ones fit. Like:
4x^2-5x-6
Factor first coefficient: (1,4), (2,2), (4,1)
Factor last coefficient: (1,6), (2,3), (3,2),(6,1)
This one works since of the answer is in the form of (ax+b)(cx+d) the first coefficient would be ac and the last coefficient would be bd. The middle one would be (ad+bc). A cross multiplication.
And you try for the mixes that work(remembering the +/-) with example, if you try (1,4) and (1,6) you would get 6+4.
The right case is (1,4)(2,-3). Which means (x+2)(4x-3).
But why did that algorithm worked?
Well, first, we are multiplying the last coefficient to the first and making the first equals to one.
We preserve that the multiplication of both would be the same.
Now we have 3 cases, in the monic form, there may be one factor that divides by the first coefficient, or none, or both.
In the first scenario, if one factor can be divided by "a", we are dividing this factor with "a", but since the other one is not divided by "a", we are multiplying the monic by "a", which means the multiplication stays the same.
In the second example, we have 4x^2+4x-15.
By the method, we have. // x^2+4-60//(x+10)(x-6). Since both are not divisible by 4, the answer by the algorithm is (4x+10)(4x-6).
Which is weird. Since the result of the multiplication is different, it results in 16x^2. So we realize that we can factor everything with 2.
So the answer is 2(2x+5)(2x-3). Which is not the original answer, but it is double. If you only want to know the zero of the function it works. But be aware of that!
So, you always have some factor in common, because of the Aritmethic Fundamental Theorem, but not all.
So, if we compare with the explanation from above, we already have the multiplication right with 10 times 6. So what happens if we multiply both by 4? You are making it 4 times larger! But what happens next? We are dividing everything by 2, which means we are dividing the result by 4 again, that is why it also works.
This one is easier to prove because 4 is a square number, but I guess we follow a similar knowledge with other compound and prime numbers.
Didn't thought about the case where both can be divided, maybe it is impossible and we can prove it (if we put the additional conditions blackened pen put on one of the first comments)
I'm sorry I'm not solving for all cases, but I hope those insight helps someone have a better understanding on the method.
It is kinda funny because it came a bit natural to me. I have had some experience with polynomial equations with fractions as a result.
Wow I needed this like a year ago. Such a good method.
i love slide and divide its so easy 🫶
Thanks this is huge relief
so if you could divide in the second factor you would just divide right?
Yes.
out of all the factoring methods, this is the best one🎉
Thank you very much you literally changed my life
How u can transfer 4 to k i.e. -6 its wrong i think bcoz its different term...Please guide me if im wrong
That's just a way to think about it. It's just multiplication; then you divide by that original 4 in the end
The method is wrong. Just take the ex: 4x^2 -28x -32
I am Indian and going to explain in easy way :
We have our equation : 4x²-5x-6
We will first take the first integer and last integer
4 × 6 = 24 (keep this in mind)
We have 5x in middle
We have to split it like that if we multiply it we get 24x² and if we add it we get 5x
So ,
4x²-8x+3x-6=0
8×3=24
-8+3=5
We take 4x common in first and then 3
4x(x-2) 3(x-2)
(4x+3) (x-2)
Hope it helps you all
Thank you so much❤😊 I don't understood 😉 the 😂method of the video🎥 but you made it😝 easier thank you❤🌹🙏 duo
@@simerpreetkaur7502 your welcome 🗿🚬
Thank you, we appreciate that you took the time to explain this to us all and it was very understandable.
@@Focusquick welcome
thx bro
Thank you so much! I just started Pre Calc and my professor tried to teach us on the A.C. method, which I just didn't quite understand
4x^2 - 5x - 6
= 4x^2 - 8x + 3x - 6
= 4x(x-2) + 3(x-2)
= (4x+3)(x-2)
This is also known as middle term factorisation.
We do this by factorising the first and third terms' numbers. I.e- 4= 2,2 ; 6 = 2,3.
Now, we need to find two factors that add to form the middle factor, here 5. We can multiply the factors or add them. Here, we are taking the 2,2 from 4 and 2 from 6 and multiplying it, making it 8. And remaining is 3. So, we are getting -8 + 3 = -5.
I don't remember the full explanation, but this was how it was done in my class haha. Hope it helps someone.
This is exactly how we do it here
@@fardiah- Good to know! Where are you from?
@@LuciferAmoyai Bangladesh. Gonna sit for my Mathemathics Alevels ( As ) this month. :)
@@fardiah- We are neighbors! Good luck for your exam bhai 💯
@@LuciferAmoyai dhonnobat ☺😇
Now this is quality content.
Better than my actual teacher who I pay to teach me lol
This was so freaking helpful
We called that the “bottoms up” method when I was in 6th grade.
Huh we called it middle term break....with some variations
Its really a bad Idea ...
Knowing the actual calculation is best.....😊
Thats the most goofiest way to factor.
The method is called "Middle Term Factorisation" if anyone interested.
This was awesome. Thank you, thank you, thank you!
My precalculus teacher was reviewing some old work for algebra and he called this method the slip and slide.
Bro confused me even more, but the method is a w
Its called mid term breaking and the way my teacher told me it is way easier than this. His explaination was simple and easy
thank very much bro to help me to solve this problem
This was the only reliable polynomial factoring strategy ive ever learnt and I will always use it.
Cool. I like the ones that are obvious once you see it done
This makes it so much easier than my math teacher tryna teach it
Very good explanation
Just a slight variation of splitting the middle term and grouping in pairs.