3 factoring tricks that you probably didn’t know

Рет қаралды 150,086

Күн бұрын

These are my top 3 polynomial factoring tricks for math Olympiad competitions. We will factor 3 quartic polynomials by "the double-cross method", "substitution", and "a hidden sum of two cubes with the difference of two squares" If you are a high school student living in the NorCal area, then consider participating in the Berkeley Math Tournament: www.ocf.berkeley.edu/~bmt/ for more info.
0:00 The "double cross" method
5:37 by substitution
7:33 Hidden the sum of two cubes and the difference of two squares
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Пікірлер: 174

@blackpenredpen Жыл бұрын

if you are a high school student living in the NorCal area, then you should definitely consider participating in the Berkeley Math Tournament on Saturday, November 5th, 2022. Please visit www.ocf.berkeley.edu/~bmt/ for more info. I will be going to BMT as a guest. We (meaning all my subscribers and myself) will be sponsoring BMT so they can offer scholarships of $1000 to the first place, $500 to the second place, and $300 to the third place. Big thanks to all my subscribers and special thanks to all my patrons and channel members. I know this is not possible without you guys! Also, Shout out to the whole BMT team (all the organizers and problem writers)! I can’t wait to meet everyone there! If you are also interested in sponsoring BMT and making an impact, please contact communications@bmt.berkeley.edu Thank you!

@leonardobarrera2816 Жыл бұрын

And for Southtowns América pearsons :(

@wowZhenek Жыл бұрын

For the second method it is arguably better to do this substitution: u = x^2 - 2x + 4, so you end up with (u-1)*(u+1) - 3 = u^2 - 4 = (u - 2)*(u + 2)

@tbg-brawlstars Жыл бұрын

Yes

@fiathla Жыл бұрын

I would have done that, too

@spikypichu Жыл бұрын

Exactly what I was thinking!

@adityaraajsingh4322 5 ай бұрын

Just use the quadratic formula 🤣

@rollingdigger19 Жыл бұрын

You really understand my needs as a calculus student. Factoring are one of most obstacles I get stuck at.

@MyOneFiftiethOfADollar Жыл бұрын

Thanks for being honest about how the problems you create are "designed" to work out nicely if one applies the correct factoring technique. Lotta Love!

@o_s-24 Жыл бұрын

Math is really satisfying when you get what's going on

@fafa47t Жыл бұрын

Thanks It's really great and your solutions are really intelligent😯 I liked the "double cross method" it's so useful👍🏼

@prefabrication 7 ай бұрын

I've always had a problem with factoring as if I can't visualize it well. The second substitution method (specifically U-substitution) is a brand-new favorite of mine. Thank you very much!

@coltonboxell1960 Жыл бұрын

The cube trick really blew my mind! Thanks

@Swiftie-d2u Жыл бұрын

for the 1st question i personally wouldve just polynomial long divide after factor theorem i didnt even know this sht is possible thxs

@spikypichu Жыл бұрын

These are some great techniques, thanks!

@reddestglizzest Жыл бұрын

Your expressions being all proud about Makinh up your own methods makes me so happy😭I truly wish to achieve that soecific achievement

@Enormousguy Жыл бұрын

Thanks for teaching me this amazing method

@owlsmath Жыл бұрын

That was great! I like the third problem best :)

@reidflemingworldstoughestm1394 Жыл бұрын

Very nice. Finding on your own is the best.

@SuryaBudimansyah Жыл бұрын

Good luck with the tourney, may the best wins!!

@SuperYoonHo Жыл бұрын

Thank you sir!!!

@chadd9907 Жыл бұрын

thanks teacher this was very nice i am impressed especially last one.

@tedkaczynski2727 Жыл бұрын

in the second one i considered to use u =x²-2x+4 so we will have (u+1)(u-1)-3 and we will have u²-2² :)) and then we will have (x²-2x+2)(x²-2x+6) easily and fast

@johnnykrocker5604 Жыл бұрын

why, we will have u^2-2^2?

@gdtargetvn2418 Жыл бұрын

@@johnnykrocker5604 yes, that's the point

@aayushswami3022 Жыл бұрын

Nice

@Arbion26 Жыл бұрын

@@johnnykrocker5604 because after we factor we'll have u^2-1-3 Simplify u^2-4 Rewrite 4 as 2^2 u^2-2^2 Now you can factor (u-2)(u+2) Now you should resubtitute, but im to lazy to do that

@johnnykrocker5604 Жыл бұрын

@@Arbion26 ohh, yes! I already see this. Thanks bro.

@Fraud_watch Жыл бұрын

Nice video as usual

@bryanjara477 Жыл бұрын

Mr. Blackpenredpen I really enjoy your videos and it blows my mind how you solve things, it’s just awesome!! Do you have any resource for me to practice this types of factorization? Greetings for Costa Rica 🇨🇷

@dosedemaths Жыл бұрын

@pragyapathak3918 Жыл бұрын

really great sir 🙏🏻😊

@christopertianzon3232 Жыл бұрын

thank you sir!

@EulersEye 2 ай бұрын

We are proud of you for figuring that out bro

@giselleformon Жыл бұрын

Thank you sirrr :)

@davidemasi__ Жыл бұрын

Very useful video 😁

@comingshoon2717 Жыл бұрын

GREAT!!! Saludos desde Chile

@subodhmeena9918 Жыл бұрын

Great video...i liked it 👍👍

@finrodf8442 Жыл бұрын

for the second you can use difference of squares

@fabiangn8022 Жыл бұрын

Buen video.👏🏽

@Madtrack Жыл бұрын

Ty man. Ur top tier

@bananaman4705 Жыл бұрын

This video is very powerful

@timeonly1401 Жыл бұрын

For #3: I also made the sum of cubes & factored, x^3+8 = (x+2)(x^2-2x+4) = (x+2)*u, where u = x^2 - 2x + 4. Then I rewrote the squared quadratic factor terms of u: (x^2-2x+5)^2 = [(x^2-2x+4) + 1]^2 = (u+1)^2 = u^2 + 2u + 1. The expression becomes: (x+2)u + (u^2 + 2u + 1) -1 = (x+2)u + u^2 + 2u = u ( u + x + 4) = (x^2 - 2x + 4)(x^2 - x + 8). Done!

@samueljehanno 8 ай бұрын

Wow

@jin_cotl Ай бұрын

How did (x+2)u + u^2 + 2u = u (u +x+ 4) ?? Shouldn’t it be u(u^2 + 2u + x + 2) ??

@jin_cotl Ай бұрын

I think I got it, yes the answer is correct, steps are right too but it skipped like 3 steps so I got lost.

@MathwithMarker Жыл бұрын

That was perfect

@user-yk1lz7gb2t Ай бұрын

Very useful content

@SidneiMV 3 ай бұрын

Nice tricks! Factorization is an art.

@jin_cotl Ай бұрын

I agree

@niom9446 Жыл бұрын

Fantastic

@holyshit922 Жыл бұрын

x^4-4x^3+12x^2-16x-15=0 Firstly group the terms with x^4 and x^3 and rest (x^4-4x^3) -(-12x^2+16x-15)=0 Now we complete the square the expression in leftmost bracket (x^4-4x^3+4x^2)-(-8x^2+16x-15)=0 (x^2-2x)^2-(-8x^2+16x-15)=0 Expression in the other bracket is quadratic and will be perfect square when its discriminant is equal to zero If we try to calculate discriminant now it may appear that discriminant is not equal to zero so we have to introduce parameter to make discriminant dependent on it We introduce parameter in such way that expression in leftmost bracket remains perfect square (x^2-2x+y/2)^2-((y-8)x^2+(--2y+16)x+y^2/4-15)=0 Calculate discriminant and force it to be equal to zero (-2y+16)^2-4(y^2/4-15)(y-8)=0 (y-8)(y^2-60)-4(y-8)(y-8)=0 Cubic resolvent is partially factored and is easy to see that y=8 is solution (x^2-2x+4)^2-1=0 (x^2-2x+3)(x^2-2x+5)=0 x_{1}=1-sqrt(2)i x_{2}=1+sqrt(2)i x_{3}=1-2i x_{4}=1+2i This method for quartic is quite easy and never fails but usually needs to solve cubic equation (so called cubic resolvent)

@hazelqawamid 8 ай бұрын

He's really happy because he made those equations 😭💗

@w__a__l__e Жыл бұрын

whats annoying is ive been subbed for a while but literally never see your work in my feed.. my brain has been missing this

@KazACWizard Жыл бұрын

First one you can just express it as f(x) and set it equal to the product of two quadratics. Expand coefficients and equate them to f(x)

@DaMoNarch91 Жыл бұрын

Not as quickly for a math tournament

@KazACWizard Жыл бұрын

@@DaMoNarch91 it takes 2min

@MyOneFiftiethOfADollar Жыл бұрын

Why couldn't you "just express it" as the product of a linear and a cubic? That requires some explanation.

@alham9656 Жыл бұрын

For the first method, does the answer need to end up being (u +- C1)(u +- C2) where c1,c2 are just constants? Ive tried workijg backwords with the u's not being equal and it doesnt really work out well, but for every one of tried in the form of yours it seems to work !

@Bplayer1113 Жыл бұрын

Hello, I am new subscriber. I saw your graduation picture and was wondering at what age did you graduate from UC Barkley😊

@brendanconnelly1075 Жыл бұрын

I run the math club at Acalanes high school in lafayette. We will be there!

@blackpenredpen Жыл бұрын

Excellent!!! See you there!!!

@BraddersVee Жыл бұрын

For Q1, how do you know to choose 5 and 3 rather than 1 and 15 ?

@robertpearce8394 Жыл бұрын

As the video states, it is a guess but an educated guess as 5,3 looks mote likely than 15,1. You can try 15,1 but the answer will be incorrect and time is wasted as this is a competition. If you apply this method to problem 3 you have choices of 32,1 or 16,2 or 8,4. I tried 8,4 first as it seemed most likely.

@numericalcode Жыл бұрын

Props for the donation!

@blackpenredpen Жыл бұрын

Thank you.

@greensalad_1205 7 ай бұрын

Can I do the cross method for cubic formula?

@charlesgoodson5774 2 ай бұрын

Sir, I was wondering, if you have a quartic polynomial with integer coefficients, is it always factorable into the product of two quadratic polynomials (even if all roots are complex OR irrational real)? If so, will the coefficients in those quadratics be integers necessarily or might they be non-integer rational numbers?

@NarenderKumar-ll6xg Жыл бұрын

Sir find the factor of the equation X ^ 4 - 4 x cube + 8 x square minus 8 x + 4 equal to zero

@FreemonSandlewould Жыл бұрын

Regarding the BMT are the scores race adjusted? Or is it fair where ONLY the work counts?

@blackpenredpen Жыл бұрын

I am not sure how they will score this time. You can find their contact info in the description

@michaelbaum6796 11 ай бұрын

Great👍

@davidbrisbane7206 Жыл бұрын

I knew someday we'd be double crossed by blackpenredpen.

@dandark5554 Жыл бұрын

I'm sorry to be stupid but i don't get one thing: Why didn't you consider 15 as 1x15 in the first method and went straight on 3x5?

@Sampreet. Жыл бұрын

I guess one clue would be to look at the coefficients of the terms. They all have factors of 2 or 3 and 2 + 3 = 5 which is a factor of the constant term. These patterns suggest that it might be easier if we were to proceed with 3 * 5 instead of 1 * 15.

@jin_cotl Ай бұрын

10:57 how did one (x^2-2x+4) go away? I’m so confused

@kennyfsTW Жыл бұрын

try x^15+1, it's very tricky but fun.

@blackpenredpen Жыл бұрын

Interesting, what's your final answer?

@blackpenredpen Жыл бұрын

WFA gives this: www.wolframalpha.com/input?i=factor+x%5E15%2B1

@kennyfsTW Жыл бұрын

@@blackpenredpen Just as WFA gives it. The point is that extracting x²-x+1 requires finding the roots of x¹⁰-x⁵+1.

@hardeepsinghg13 Жыл бұрын

Will you conduct any online competition in future for students not in your country? Like India Pls try doing it some day, for Grade 12 students

@blackpenredpen Жыл бұрын

There’s a BMT online. U can check that out. 😃

@jaii5955 Жыл бұрын

Double cross method is greatest method ever

@cathaywongkt Жыл бұрын

Was a square sign missed in the 3rd (blue) expression in the thumbnail? (Sorry not sure if it is mentioned).

@blackpenredpen Жыл бұрын

You are right! thank you.

@EyeSooGuy Жыл бұрын

Hey there. Can you explain to us what tetration, pentation, hexation, etc is? 😈

@alejrandom6592 Жыл бұрын

What happened to the "main dish" video?

@enjoywatchingyoutube3346 Жыл бұрын

2:33 shoudnt that be -2 and -3 ? cuz X1+X2=-p ?

@mcguides9176 Жыл бұрын

-2 and -3 are the roots but this is dealing with factorizing the polynomial is (x-(-2))(x-(-3)) so when the x coefficient is positive the roots are negative but when you factorize it the negative negative becomes a positive

@ahmadsalehin8329 Жыл бұрын

whoa i did this individully at home!!! 2:50

@tomshimon Жыл бұрын

In the first method/problem how did you know how to split the 15?

@jin_cotl Ай бұрын

Basically, never use 1 times 15 unless there is no other choice like 1 times 3 or something.

@charlesbromberick4247 Жыл бұрын

Mr. Burp is an absolute whiz!

@user-rx3iv7bl6b Жыл бұрын

Why we factored 15 as 3 into 5 not 15 and 1

@sajjadali-taeminsteppedony8434 Жыл бұрын

Equation zaddy is back 😶

@noelyvalisoarakotoarison7240 Жыл бұрын

Splendid, Thanks.. Can these methods become standard for resolving quartic polynomial

@dosedemaths Жыл бұрын

@jin_cotl Ай бұрын

Absolutly

@garyhuntress6871 Жыл бұрын

my patreon halted because my cc expired. I'm going to fix that today!

@tsaqifrizky5276 Жыл бұрын

Imagine facing bprp in a math tourney you'd get smoked 10 different ways (with an elegant proof for each)

@crazybanana0272 Жыл бұрын

All the expo markers in the bottom right corner of the screen waiting for their turn to be used

@user-et1up1nk9k Жыл бұрын

🙂

@vishalmishra3046 Жыл бұрын

*First Technique is not Generalizable* The first technique ONLY works when you predict the correct factors of the constant term. What if it is not an integer in the equation. What if its factors are not integers (e.g. 15 = 9 x 5/3). If the constant term has many factors (large N), then you will discover these issues after a very large number of attempts (Number of pairs of factors = N x (N-1) / 2 attempts). And then the same number of attempts with negative factors (e.g. 15 = -3 x -5). That's a lot of factor-pairs attempted just to eventually find that this technique will not work.

@MyOneFiftiethOfADollar Жыл бұрын

Excellent point which detracts from the viability of the technique, especially for large constant terms.

@CAFoundationExam Жыл бұрын

@muhammedrehaan5750 Жыл бұрын

Thank u sir but my brain popped😅

@dosedemaths Жыл бұрын

@user-zt6ds2cb2j 11 ай бұрын

Is it true: The method which is applicable to factor a 4 degree polynomial is depend upon the way i represent it to the world.🎉🎉

@jrsun Жыл бұрын

what did you mean by factoring by squares

@dosedemaths Жыл бұрын

hand-pink-waving

@dosedemaths Жыл бұрын

@jin_cotl Ай бұрын

Breaking down squares into smaller multiples

@killing_gaming0973 Жыл бұрын

I will also say hi can cheer you up too

@glinzaglinza7917 Жыл бұрын

Please do some lebesgue integrals

@hydropage2855 Жыл бұрын

NO, PLEASE DONT

@glinzaglinza7917 Жыл бұрын

@@hydropage2855 haha just don't watch it 😅

@afuyeas9914 Жыл бұрын

The chad answer: Ferrari's method every single time

@Happy_Abe Жыл бұрын

How’d we know the first one wasn’t a product of cubic and linear instead of two quadratics

@ultimatedude5686 Жыл бұрын

Any cubic can be factored as a linear term and a quadratic term. If you have a linear term times a cubic, you can convert that to two linear terms times a quadratic. If you multiply the linear terms you get the original polynomial factored in terms of two quadratics. Therefore, all quartics can be factored as two quadratics.

@Happy_Abe Жыл бұрын

@@ultimatedude5686 that’s true yeah thanks! This is because all cubics have at least one real solution I forgot about that

@lucasmontec Жыл бұрын

The cross method gives you a right result... This means that the double-cross method actually screws you up? (Haha)

@youssefbenmMorocco Жыл бұрын

From Morocco 🇲🇦🇲🇦🇲🇦 I fellow you just to understand how est stupids were somes of my math's teachers🤣🤣🤣 Did u come to morocco?!

@noobplayer5461 Жыл бұрын

These are olympiad question!!!!😂

@dosedemaths Жыл бұрын

@wrestling_is_fake Жыл бұрын

Request: how are trigonometric values of angles beyond 90° found?

@olli3686 Жыл бұрын

"because i made this" LOL

@broytingaravsol Жыл бұрын

有名次有錢拿哦

@blackpenredpen Жыл бұрын

對

@nuctang Жыл бұрын

How about x^100+1? Lol.

@RegisteredLate123 Жыл бұрын

ooorrrrr just try and guess one solution then long divide?

@mahhar31 Жыл бұрын

I am from India

@user-et1up1nk9k Жыл бұрын

I am not from India

@chessematics Жыл бұрын

Me too but no one asked bro

@guywhoasked6046 Жыл бұрын

@@chessematics I did 🗿

@Aditya-zs8hz Жыл бұрын

Pucha kisi ne

@mahhar31 Жыл бұрын

@@user-et1up1nk9k Where are you from?

@Dylan-oq8xb Жыл бұрын

I think basic factor theorem would be a lot more straight forward 😂

@blackpenredpen Жыл бұрын

What’s basic factor theorem?

@j.u.4.n620 Жыл бұрын

😂

@owlsmath Жыл бұрын

I think this is just guess and check for zeroes right? Then you can reduce the degree of the polynomial.

@Dylan-oq8xb Жыл бұрын

@@owlsmath yeh I just couldn’t be bothered to explain it 😂

@owlsmath Жыл бұрын

@@Dylan-oq8xb I googled factor theorem and then it all made sense 😃

@richardbraakman7469 Жыл бұрын

Missed chance to use fibonacci numbers for the prize money :)

@dosedemaths Жыл бұрын

🍉

@user-fe3ww3hh3s Жыл бұрын

中文亂入,看無,看無。前三名有錢?真是不好。賽馬? 鼓勵童學參加挑戰,因資金有限只能提供前三車馬費,會不會比較好。另,比的是解題速度?解題邏輯與方法無法評斷?舉個不可能例子,如果有學生把簡單題寫成像愛因斯坦般申論題如何辦,一張紙不夠寫?

@blackpenredpen Жыл бұрын

歡迎一起來贊助喔這樣你要怎樣分配都可以