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linear polynomials with the same variable. As in (mx + c)(nx +d), when you FOIL it becomes (mn)x^2 + (dm)x + (cn)x + cd. Which we then simplify to (mn)x^2 + (dm + cn)x + cd. At leazt for me this is the only time I was told to FOIL. Idk what education is elsewhere but otherwise you were told to do each-by-each. NEVER seen someone use foil in a scenario that wasnt the factorisation of that form. (By the way x could be something like y^2 so it isnt strictly linear polynomials but that scenario is EFFECTIVELY the same and has the same upsides)
I realise I didnt make it clear but FOIL orders the products in the way you want them to. While each by each usually adds a step of finding where the like terms are and reording stuff in order of powers of x/y/etc.
@@TheKastellan does that really matter? either way you end up with the same products. you really only have to remember the signs and stuff like exponents and roots etc
Yup! For anyone who didn't learn this in school, the S stands for 'second' and T for 'third. Happy FOIL(FS)(FT)(SF)(SS)(ST)(SL)(TF)(TS)(TT)(TL)(LS)(LT)ing!
My younger sister was taught FOIL (in australia) and struggled with expanding polynomials bc it's so easy to literally just forget what FOIL means and what it wants you to do. She was never really taught what it actually means to expand polynomials, only how to follow a weird and confusing set of instructions. Teaching FOIL is the opposite of teaching math
@@asagiai4965 pretty much, except the distributive property is the general form. The distributive property states that a(b + c) = ab + ac. Therefore you take (a + b)(c + d) = (a + b)c + (a + b)d = ac + bc + ad + bd. But, if you want to be pendantic, you can also generalise it for: (a + b + c)(d + e + f) = (a + b + c)((d + e) + f) = (a + b + c)(d + e) + (a + b + c)f = (a + b + c)d + (a + b + c)e + ((a + b) + c)f = ((a + b) + c)d + ((a + b) + c)e + (a + b)f + cf = (a + b)d + cd + (a + b)e + ce + af + bf + cf = ad + bd + cd + ae + be + ce + af + bf + cf. Or you could use general distributive property like: (a + b + c)(d + e + f) = ad +ae + af + bd + be + bf + cd + ce + cf...
@@asagiai4965 it literally is. its just the distributive property with extra (unnecessary) steps. it helps better to learn "multiply each term with each of the other" because thats literally it TwT
@Beheldtothesun yes do each by each is correct, but the reason foil exist (despite being similar) is probably because it helps you think Which part of each by each are you now. In other words it helps you remember which two terms you are multiplying. So I think knowing each by each is much better. But knowing foil is not bad also
@asagiai4965 not necessarily. say if im to do (a + b + c) (d + e + f) i can just write it out as d(a + b +c) + e(a + b + c) + f(a + b + c) if i need it to be more clear even so, its not necessary to do that if you just mark under each term (like just underline the term or circle it or something) you're done with multiplying and it does the exact same thing edit: spelling error
Odd. I never learned anything like FOIL in my school. What we learned was that you can distribute a number across terms in a parenthesy and that you can treat an entire parenthesy as a unit. So when it came to multiplying two parentheses, we just distributed one of the parentheses across the terms of the other and then distributed the terms of the other over the terms of the first.
The truly enlightened foiler first realizes a = (a + 0)(1 + 0) b = (b + 0)(1 + 0) c = (c + 0)(1 + 0) so a(b + c) = [(a + 0)(1 + 0)][(b + 0)(1 + 0)+(c + 0)(1 + 0)] which we can simplify to [(a + 0)(1 + 0) + (0 + 0)(0 + 0)][(b + 0)(1 + 0) + (c + 0)(1 + 0)] And then from there it's pretty straightforward, though if the student is still struggling with conceptualizing the value of 1 in this equation, it can be helpful to think of 1 as (1 + 0)(1 + 0) just to clean things up a little bit
To be honest I had a major math break through when I worked out that long multiplication is simply a type of polynomial multiplication. (21)(43) = (2*10 + 1)(4*10 + 3)
My favorite "mathism" are what i like to call "math poems." Im sure youve heard one before. For example "the derivative of the sum is the sum of the derivatives... the limit of the product is the product of the limits..." etc. I dunno, your intro made me think of that.
This isn't an axiom, but it is a poem you might appreciate. [(12 + 144 + 20) + 3sqrt(4)]/7 + 5*11 = 9^2 + 0 a dozen a gross and a score plus three times the square root of four divided by seven plus five times eleven is nine squared and not a bit more
😂 When I was in college and helping a math teacher out in her 8th grade class, she wanted me to help the kids as they worked on long division, at which point I realized I never learned long division the way most Americans do. Felt like a real dumbo!
I am sympathetic to that perspective. I respect the utility of them though. Even so much as, in a classroom setting, completely committing to overusing a mnemonic can help the students remember just by having all these key moments in class that the phrase keys their brain into.
Learning should be about knowing when and how to use a tool correctly instead of trying to make students recall what the tools even are. Give them the tools and teach them to use them if you truly care about students learning and not just pointlessly memorizing.
I agreed with the thing that foil is stupid in the sense that it hinders learning the actual concept of polynomial distribution. It is taught as a method for less mathematically inclined students to just be able to solve the problem without actually understand the concept. And that is ok to an extent because at least they are learning something. At my school we were taught both the formal method and foil as an emergency pneumonic in case we blank on a test. You have to think of your self in the shoes of a 8th or 9th harder just first learning algebra. It may seem so simple now but back then these things were very helpful. Eventually all students continuing through math will learn the concept of distribution and they go up into algebra 2, precalc, and calc. Would it be nice if schools always taught the proper wat? Yes, however they have to cater to the masses who may still struggle with fractions in 9th grade. Concepts like binomial expansion working upon this was dropped in us curriculums because they were “too hard” for them. Back to the video, your table method is even stupider than foil. At least foil is foolproof and quick for binomial x binomial which is the most common setup. The table method is slow and teaches even less about the concept of distribution. Also you spent a good chunk of your video distributing constants over other constants. Foil and distribution in general was never taught for this stupid application. Everyone would just add and multiply. What people have to realize is (a +b + c+…)(d+e+f+…) is the same as a((d+e+f+…)+b((d+e+f+…)+c((d+e+f+…)+…((d+e+f+…) In conclusion stupid video 2/10 two points for having a correct opinion on foil.
1. I respect the utility of FOIL and mnemonics in general. 2. I don't like the table method either. If you watched the video you might have heard me say 'that was nice and easy, we just had to make this giant fucking table', which was my way of saying 'I don't like this method either'. 3. Constants? Most of the video was done with letters which were never specified to be constants or variables. 4. *mnemonic In conclusion, stupid comment.
I think the better way to deal with polynomial multiplication is thus: Let N be the number of terms in the first, let M be the number of terms in the second. Draw a grid of size NxM Write the terms of the first polynomial along the top, above, of the grid, write the terms of second along the right, beside, the grid. For every cell in the grid, multiply the terms to the top and to the right outside the grid. Add up all the terms in the grid Done. a....b ac.bc.c ad.bd.d ac+bc+ad+bd
The "table method" was always called the "box method" in my education, and I'm its most arduent defender anytime someone brings up FOIL. It has the added benefit of making the 3blue1brown video on convolutions feel more satisfying.
this is fantastic, but i'm honestly distracted by how consistently thick your sharpies are. do you ever get duds that are thin?? these are all fantastically bold
FOIL was one method that I remember being taught in school. But I never use it because it's pretty confusing in my opinion. One of my math teachers also taught the table method. Which is what I prefer to use, because it's easy to setup and use compared to FOIL. Even the math teacher who taught it prefers this method.
As to mnemonics, my personal wrath of math is aimed at PEMDAS. I think that most middle schoolers would grasp the concept of a field (using Q as an example). Having that, one could replace a division sign b with a *(1 / b) and proceed with the usual rules for multiplying and adding in a field. Once, long ago, when I was in university, I asked a couple of my non math major friends if they'd ever heard of the distributive law. That led to some teaching moments. Based on your presentation, I think I could write an inductive proof where I claim to prove (in the spirit of Euclid) that the Foil Method May Be Extended Indefinitely. I can't see any reason why anyone would want to do that. Interestingly enough, Euclid never said that he attempted to prove that there were infinitely many primes. He said that primes could be produced "indefinitely" which amounts to the same thing, but infinity itself was never on Euclid's menu.
something that my 7th grade math teachers taught was instead of PEMDAS was GEMA standing for: Groupings (parentheses, fractions, so on), Exponents, Multiplication (and division since you can multiply by a reciprocal), and Addition (and subtraction since you can add a negative)
What’s so interesting about Euclid’s word choice? The statements “there are infinitely many primes” and “primes can be produced indefinitely” are logically equivalent.
I vaguely remember this being mentioned in school, but I barely paid any attention in math class and just did my own thing since I learn best when I see the material and then figure out why that is for myself. So I proved all these things on my own and found my own methods to do things. Teachers didn’t like that I didn’t show my work but I always got the right answer and never cheated so they kinda just had to deal with it.
The only time a use a semi foil method is when multiplying arbitrary polynomials of arbitrary degree in a polynomial ring. It helps group terms with same degree. (Also for formal power series where an infinite table might be cumbersome)
When you learn foil, you're learning about binomial expressions. So you use a method for binomials. When you learn about expressions including more terms, then you can learn another method. The best method is the one that works for whatever specific application you're dealing with. Generality is not inherently desirable. No one who learns foil is ever going to use it to solve your example with the literals.
That's fair but it leaves many students confused when they encounter things beyond binomials because all they know is the FOIL magic trick; when if they had just learned that to multiply binomials they use the distributive property (which they already know), they wouldn't even need to be taught how to deal with bigger examples, they would just know how to apply properties of mathematics they've mastered before even to new situations.
@WrathofMath Of course not. When they are introduced to a new concept, they're also introduced to a new method for handling the concept. Just like when you learn any new concept in math.
I’ve encountered many students who are confused by concepts more basic than expanding polynomials. When they get FOIL, it gives them a boost of confidence that they need to keep going. There’s plenty of time to teach them each by each or the grid method as they progress.
Foil technically does give you how to multiply any polynomial by any polynomial. Simply bracket the sum inside the polynomial so that there's only one main addition, then foil, remove the brackets, and repeat. It thus technically is all you need for any n-degree polynomial multiplication for example: (a+b+c)*(d+e) = (a+(b+c))*(d+e) = ad + ae + (b+c)*d + (b+c)*e = ad + ae + bd + cd + be + ce Edit: lol I commented before watching the full video! Glad this was addressed.
Foil is silly yes, but it is a good first method of learning expanding these things out, I was taught distribution to each, then the area method as a easier way to write out long ones, and foil for those smaller expands. Foil is helpful for teaching beginners just a quick method
10:10 I use a method inspired by base 10 number column multiplication and omit the x to the power just like omitting 10 to the power in the column multiplication. So treating x as a “base” kinda
I think I tend to treat FOIL as a sort of "canonical ordering" when expending products of two binomials, and then extend that canonical ordering to more complex products, giving one less thing to think about. Said canonical ordering by the way does not match the given "mnemonics" for trinomial and qutranomial products. Rather, it's the first term on the left times everything in the right, plus the second term on the left times everything in the right, etc. Only after expanding each pair of terms do I collect like terms. Which stage I normalise the ordering of different variable factors depends on whether the product is commutative or not.
I was thinking, it might just be easier, assuming distributing a product over a sum is introduced first, to just say "Hey, this left sum is also... just a term itself we can distribute. So we get a sum of these products, then we do normal distribution over each sum term, and just add them all together. It doesn't introduce any new ideas or require any new mnemonics, it just re-uses old ideas. Then, once students are able to compute expressions like that, maybe introduce some shortcuts to make it easier.
I guess my math textbooks were remiss because I missed this mnemonic in my education. Why do you even need a mnemonic to distribute the terms? Sure, it's easy to miss one, or several, but that's not because you didn't remember what to do, you just did it sloppily. There is no need for a pneumonic here. Now if there was one for _factoring_ a polynomial, I'd love that (there probably is. I clearly missed the chapter on mnemonics lol)
The moment i learnt about variables, coefficients, terms, and polynomials, i immeadiately knew to multiply two binomials i just use the distributive property. I don't think you should just teach people foil and abstract it that way, you should let them know how the distributive property applies to polynomials and it might help with even multiplying trinomials or bigger polynomials.
FOIL is fine, it's a concrete example of a general concept to be studied later. "Let's reexamine FOIL in a more general way..." can't be uttered until you've learned it
Funnily enough I had never even used the term “foil” used before a week ago, I just always followed the multiplication to its logical conclusion, as I was homeschooled for most of my education and it seems to have, for the most part, been beneficial for my learning, I would never just learn a method to do it without thinking using rule such as “foil” ever like it was a spell, I would much rather truly learn the logic behind it creating a far better intuition for things like math as it serves much better for everything in life. There my essay is done(sry for rambling)
I must have missed the day “foil” was explained because I was literally 27 years old when I learned it was an acronym haha. I just thought it was a weird bit of school math jargon and always felt that “unfoil” would have made more sense.
I’ve always been partial to the “ box method” which seems very similar to your tabular method, except instead of multiplying, just the coefficient you multiply with the coefficient and the given power of X (it actually works for any set of variables as well, such as trying to multiply (ax by) and (cx^2 fy^3) for example). This means you don’t have to include all the tedious zeros in your tubular method, but still allows the handling much larger polynomials. ua-cam.com/users/shortsYOUzwI6wX_s?si=HTofRxiFtpCF4HfM this is a good video demonstration of it.
A generalization of multiplying any nxnomial by any mxnomial is to do a dot product of the two equations. You will always get the correct answer by doing this.
@@isaiah0xA455 A simple linear equation, the coefficients can be expressed as a vector or matrix. Taking the dot product of the coefficients is the same as what is shown in this video
@@krwada The dot product only multiplies like terms by like terms. It would leave out the “outside, inside” part of the FOIL algorithm. It doesn’t resemble the table multiplication at all.
No it wouldn’t have, it is very slow. Just distribute each term of the first polynomial to each term of the second polynomial. This guy is complaining about foil and proposes an even worse method.
I don't like the table method either, and I denigrate it in the video. But my role here is to hate on FOIL, so if someone prefers a different method I fully support that
Unless you chose to study maths further than you are required to, people who learn FOIL in school are only ever going to need to multiply binomials, not trininomials, and you only need common sense to know not to use it on (2+3)(4+5) because when you are taught FOIL you are taught it for when you CANT combine the terms in the brackets, such as (2x + 3) If you do study maths as your chosen subject, then you’re probably smart enough to know to disregard FOIL as you move into any polynomial multiplication more complicated than binomials
I think that if you accept any Mnemonics, FOIL isn’t a bad one compared to many others. If it’s presented carefully, the “FOIL Method” can actually help students remember how to use distributivity of multiplication over addition. Your concern about using this method to compute products of numbers that are written as sums also is, in my opinion, a bit overblown, especially with that clickbait thumbnail that suggests an incrimination towards those who teach using that particular Mnemonic. Consider the problem of multiplying two matrix binomials: (A+B)(C+D)=AC+AD+BC+BD. Did I use the “FOIL” mnemonic to do that? Yes, but I didn’t write it out. Writing the following would have been wrong, however: (A+B)(C+D)=CA+AD+CB+BD. Of course, if you know, you know, that matrix multiplication isn’t commutative. A problem with teaching binomial expansions using the “FOIL” mnemonic, teachers shouldn’t apply commutativity in a first step, only because not all students are going to never take linear algebra or matrix algebra, or some other form of Noncommutative algebra. The kind of example that you exhibited involves numerals that are small in value, so that certain useful mental arithmetic tools aren’t well illustrated, but that’s exactly one I’d the reasons the “FOIL” mnemonic is taught. To review that scenario (with small numbers again but not the same), we see that a more detailed illustration would be the following “proof” that a product of 7 with 8 is 56: (5+2)(5+3)=(5+2)(5)+(5+2)(3)=(5)(5)+(2)(5)+(5)(3)+(2)(3)=25+10+15+6=35+15+6=56. Now, if you teach a second grader to do arithmetic this way on certain problems, they learn how the properties of arithmetic work for the step by step justifications. Then during a parent teacher conference you’ll be accosted by parents who already can multiply pairs of one digit numbers whose product odd a two digit result, but their second grader isn’t there yet. Why would you be accosted (verbally)? Well, when the child did their homework, they asked for help from a parent who isn’t equipped to explain the steps or how the distributive law applies in this case. If someone knows how to justify each step above, then they can also justify some useful things about products of larger numbers. For instance, consider the problem of multiplying 435 by 263: (435)(263)=(400+35)(203+60) =(400)(203+60)+(35)(203+60) =(400)(203)+(400)(60)+(35)(203)+(35)(60) =(4)(10^2)(200+3)+24(10^3)+(35)(200+3)+(30+5)(60) =(4)(10^2)(200)+(4)(10^2)(3)+24(10^3)+(35)(200)+(35)(3)+(30)(60)+(5)(60) =(8)(10^2)(10^2)+(12)(10^2)+24(10^3)+(70)(10^2)+(30+5)(3)+(18)(100)+(30)(10) =24(10^3)+(8)(10^4)+(12)(10^2)+(7)(10^3)+90+15+(18)(10^2)+(3)(10^2) =8(10^4)+(24+7)(10^3)+(12+18+3)(10^2)+90+15 =11(10^4)+(1+3)(10^3)+3(10^2)+100+5 =114405. This is by far a not most efficient solution, and one would not use it to solve problems quickly, but to illustrate how the laws of arithmetic work so you can teach yourself some speed for mental arithmetic. Doing no problems in this much detail would make the mental math tricks look more like magic, and so a healthy mixture of methods is better. Here’s a way to use the “FOIL” method on the above problem instead: (435)(263)=(500-65)(200+63) =(500)(200)+(500)(63)-(65)(200)-(65)(63) =(5)(2)(10^4)+(5)(63)(10^2)-(130)(10^2)-(60+5)(70-7) =(10^5)+(315)(10^2)-(13)(10^3)-((60)(70)+(5)(70)-(60)(7)-35) =(10^5)+3(10^4)+(5)(10^2)-(13-1)(10^3)-((42)(10^2)+(35)(10)-(42)(10)-35) =(10^5)+3(10^4)-(13-1)(10^3)-(42-5)(10^2)+(42-35)(10)+35 =(10^5)+3(10^4)-12(10^3)-37(10^2)+(7+3)(10)+5 =(10^5)+(30-12)(10^3)-(37-1)(10^2)+5 =(10^5)+18(10^3)-36(10^2)+5 =(10^5)+(10^4)+(80-36)(10^2)+5 =(10^5)+(10^4)+(50-6)(10^2)+5 =114405. Needless to say there are much faster ways to compute this, but the “FOIL” Mnemonic can help one do all the above steps in one’s head, rather than writing it all on paper. Shortcuts are needed for students who participate in arithmetic competitions. Not knowing the “FOIL” method can in some cases cost a student precious time in such an enjoyable competition. Those who object to showing the above steps may in many cases be those who’d rather you just do a calculator problem instead. Of course, some computations are competitions, but some are not. Check the steps: (435)(263)-(400+35)(203+60)=0 (400+35)(203+60)-((400)(203+60)+(35)(203+60))=0 (400)(203)+(400)(60)+(35)(203)+(35)(60)-((4)(10^2)(200+3)+24(10^3)+(35)(200+3)+(30+5)(60))=0 =(4)(10^2)(200+3)+24(10^3)+(35)(200+3)+(30+5)(60)-((4)(10^2)(200)+(4)(10^2)(3)+24(10^3)+(35)(200)+(35)(3)+(30)(60)+(5)(60))=0 (4)(10^2)(200)+(4)(10^2)(3)+24(10^3)+(35)(200)+(35)(3)+(30)(60)+(5)(60)-((8)(10^2)(10^2)+(12)(10^2)+24(10^3)+(70)(10^2)+(30+5)(3)+(18)(100)+(30)(10))=0 (8)(10^2)(10^2)+(12)(10^2)+24(10^3)+(70)(10^2)+(30+5)(3)+(18)(100)+(30)(10)-(24(10^3)+(8)(10^4)+(12)(10^2)+(7)(10^3)+90+15+(18)(10^2)+(3)(10^2))=0 24(10^3)+(8)(10^4)+(12)(10^2)+(7)(10^3)+90+15+(18)(10^2)+(3)(10^2)-(8(10^4)+(24+7)(10^3)+(12+18+3)(10^2)+90+15)=0 8(10^4)+(24+7)(10^3)+(12+18+3)(10^2)+90+15-(8(10^4)+(30+1)(10^3)+(30+3)(10^2)+100+5)=0 8(10^4)+(24+7)(10^3)+(12+18+3)(10^2)+90+15-(11(10^4)+(1+3)(10^3)+3(10^2)+100+5)=0 11(10^4)+(1+3)(10^3)+3(10^2)+100+5-114405=0 +++++++* (500)(200)+(500)(63)-(65)(200)-(65)(63)-((5)(2)(10^4)+(5)(63)(10^2)-(130)(10^2)-(60+5)(70-7))=0 (5)(2)(10^4)+(5)(63)(10^2)-(130)(10^2)-(60+5)(70-7)-((10^5)+(315)(10^2)-(13)(10^3)-((60)(70)+(5)(70)-(60)(7)-35))=0 (10^5)+(315)(10^2)-(13)(10^3)-((60)(70)+(5)(70)-(60)(7)-35)-((10^5)+3(10^4)+(5)(10^2)-(13-1)(10^3)-((42)(10^2)+(35)(10)-(42)(10)-35))=0 (10^5)+3(10^4)+(5)(10^2)-(13-1)(10^3)-((42)(10^2)+(35)(10)-(42)(10)-35)-((10^5)+3(10^4)-(13-1)(10^3)-(42-5)(10^2)+(42-35)(10)+35)=0 (10^5)+3(10^4)-(13-1)(10^3)-(42-5)(10^2)+(42-35)(10)+35-((10^5)+3(10^4)-12(10^3)-37(10^2)+(7+3)(10)+5)=0 (10^5)+3(10^4)-12(10^3)-37(10^2)+(7+3)(10)+5-((10^5)+(30-12)(10^3)-(37-1)(10^2)+5)=0 (10^5)+(30-12)(10^3)-(37-1)(10^2)+5-((10^5)+18(10^3)-36(10^2)+5)=0 (10^5)+18(10^3)-36(10^2)+5-((10^5)+(10^4)+(80-36)(10^2)+5)=0 (10^5)+(10^4)+(80-36)(10^2)+5-((10^5)+(10^4)+(50-6)(10^2)+5)=0 …
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Why is foil even mentioned???? Isnt "each by each" way faster to understand and way more generaly aplicable????
Way harder to forget a term if there’s a different name for each
linear polynomials with the same variable.
As in (mx + c)(nx +d), when you FOIL it becomes (mn)x^2 + (dm)x + (cn)x + cd.
Which we then simplify to (mn)x^2 + (dm + cn)x + cd.
At leazt for me this is the only time I was told to FOIL.
Idk what education is elsewhere but otherwise you were told to do each-by-each. NEVER seen someone use foil in a scenario that wasnt the factorisation of that form.
(By the way x could be something like y^2 so it isnt strictly linear polynomials but that scenario is EFFECTIVELY the same and has the same upsides)
I realise I didnt make it clear but FOIL orders the products in the way you want them to. While each by each usually adds a step of finding where the like terms are and reording stuff in order of powers of x/y/etc.
@@TheKastellanno you just do each by each the order in which you do doesn't matter
@@TheKastellan does that really matter? either way you end up with the same products. you really only have to remember the signs and stuff like exponents and roots etc
FOIL(FS)(FT)(SF)(SS)(ST)(SL)(TF)(TS)(TT)(TL)(LS)(LT)
very memorable (and easily pronounceable)
Yup! For anyone who didn't learn this in school, the S stands for 'second' and T for 'third. Happy FOIL(FS)(FT)(SF)(SS)(ST)(SL)(TF)(TS)(TT)(TL)(LS)(LT)ing!
My younger sister was taught FOIL (in australia) and struggled with expanding polynomials bc it's so easy to literally just forget what FOIL means and what it wants you to do. She was never really taught what it actually means to expand polynomials, only how to follow a weird and confusing set of instructions.
Teaching FOIL is the opposite of teaching math
Sorry to hear FOIL got her :(
it always confused me why people use foil. just learn the distributive property and it instantly follows...
Isn't FOIL a distrubutative property with name
@@asagiai4965 pretty much, except the distributive property is the general form.
The distributive property states that
a(b + c) = ab + ac.
Therefore you take
(a + b)(c + d)
= (a + b)c + (a + b)d
= ac + bc + ad + bd.
But, if you want to be pendantic, you can also generalise it for:
(a + b + c)(d + e + f)
= (a + b + c)((d + e) + f)
= (a + b + c)(d + e) + (a + b + c)f
= (a + b + c)d + (a + b + c)e + ((a + b) + c)f
= ((a + b) + c)d + ((a + b) + c)e + (a + b)f + cf
= (a + b)d + cd + (a + b)e + ce + af + bf + cf
= ad + bd + cd + ae + be + ce + af + bf + cf.
Or you could use general distributive property like:
(a + b + c)(d + e + f)
= ad +ae + af + bd + be + bf + cd + ce + cf...
@@asagiai4965 it literally is. its just the distributive property with extra (unnecessary) steps.
it helps better to learn "multiply each term with each of the other"
because thats literally it TwT
@Beheldtothesun yes do each by each is correct, but the reason foil exist (despite being similar) is probably because it helps you think
Which part of each by each are you now. In other words it helps you remember which two terms you are multiplying.
So I think knowing each by each is much better.
But knowing foil is not bad also
@asagiai4965 not necessarily. say if im to do (a + b + c) (d + e + f)
i can just write it out as
d(a + b +c) + e(a + b + c) + f(a + b + c)
if i need it to be more clear
even so, its not necessary to do that if you just mark under each term (like just underline the term or circle it or something) you're done with multiplying and it does the exact same thing
edit: spelling error
Odd. I never learned anything like FOIL in my school. What we learned was that you can distribute a number across terms in a parenthesy and that you can treat an entire parenthesy as a unit. So when it came to multiplying two parentheses, we just distributed one of the parentheses across the terms of the other and then distributed the terms of the other over the terms of the first.
Yeah, that seems pretty reasonable to me! No silly single-case rules, just a general principle!
It’s parenthesis
@@davidbailis8415 it's parentheses. British Spelling.
I clicked on this video thinking it was a video about why foils like aluminium foil are stupid, like a video by technology connections
Well I'll bet you were disappointed
A similar seemingly-useless mnemonic I’d never heard of before coming to the internet was LIATE, for integration by parts.
I've seen that too actually, it never stuck because I only ever saw it in passing and have no idea what it stands for
Yeah… “foil” is mainly for the kids who just memorize the topic and never try to understand the concept.
Now I wonder if a FOILer computes a(b+c) by doing (a+0)(b+c) when can once again bask in the light of the FOIling method.
The truly enlightened foiler first realizes
a = (a + 0)(1 + 0)
b = (b + 0)(1 + 0)
c = (c + 0)(1 + 0)
so a(b + c) = [(a + 0)(1 + 0)][(b + 0)(1 + 0)+(c + 0)(1 + 0)]
which we can simplify to
[(a + 0)(1 + 0) + (0 + 0)(0 + 0)][(b + 0)(1 + 0) + (c + 0)(1 + 0)]
And then from there it's pretty straightforward, though if the student is still struggling with conceptualizing the value of 1 in this equation, it can be helpful to think of 1 as (1 + 0)(1 + 0) just to clean things up a little bit
To be honest I had a major math break through when I worked out that long multiplication is simply a type of polynomial multiplication.
(21)(43) = (2*10 + 1)(4*10 + 3)
Bro these comments are killing me
@@theVtuberCh WOAH THANKS
2:42
Me solving right angle trig with sine or cosine rule instead of SOH CAH TOA every single time
My favorite "mathism" are what i like to call "math poems." Im sure youve heard one before. For example "the derivative of the sum is the sum of the derivatives... the limit of the product is the product of the limits..." etc. I dunno, your intro made me think of that.
This isn't an axiom, but it is a poem you might appreciate.
[(12 + 144 + 20) + 3sqrt(4)]/7 + 5*11 = 9^2 + 0
a dozen a gross and a score
plus three times the square root of four
divided by seven
plus five times eleven
is nine squared and not a bit more
10:09 the acronymist in me cant help but notice dOOef aObOOc and wonder what it stands for
I never know FOIL until I begin teaching new high school students 😂 They asked me if I used foil method to expand, I was confused
😂 When I was in college and helping a math teacher out in her 8th grade class, she wanted me to help the kids as they worked on long division, at which point I realized I never learned long division the way most Americans do. Felt like a real dumbo!
I am generally against acronyms, memory tools, and memorization in general. Memorization becomes an impediment to understanding.
I am sympathetic to that perspective. I respect the utility of them though. Even so much as, in a classroom setting, completely committing to overusing a mnemonic can help the students remember just by having all these key moments in class that the phrase keys their brain into.
Bro has never taken biology and chemistry
Learning should be about knowing when and how to use a tool correctly instead of trying to make students recall what the tools even are. Give them the tools and teach them to use them if you truly care about students learning and not just pointlessly memorizing.
As much as I agree with you, it is a necessary evil.
- Med Student
@@SomeInternetUser45 Or history for that matter.
Me, CS-brained: That's a nice data structure!
I agreed with the thing that foil is stupid in the sense that it hinders learning the actual concept of polynomial distribution.
It is taught as a method for less mathematically inclined students to just be able to solve the problem without actually understand the concept. And that is ok to an extent because at least they are learning something.
At my school we were taught both the formal method and foil as an emergency pneumonic in case we blank on a test.
You have to think of your self in the shoes of a 8th or 9th harder just first learning algebra. It may seem so simple now but back then these things were very helpful.
Eventually all students continuing through math will learn the concept of distribution and they go up into algebra 2, precalc, and calc.
Would it be nice if schools always taught the proper wat? Yes, however they have to cater to the masses who may still struggle with fractions in 9th grade. Concepts like binomial expansion working upon this was dropped in us curriculums because they were “too hard” for them.
Back to the video, your table method is even stupider than foil. At least foil is foolproof and quick for binomial x binomial which is the most common setup.
The table method is slow and teaches even less about the concept of distribution.
Also you spent a good chunk of your video distributing constants over other constants. Foil and distribution in general was never taught for this stupid application. Everyone would just add and multiply.
What people have to realize is (a +b + c+…)(d+e+f+…) is the same as a((d+e+f+…)+b((d+e+f+…)+c((d+e+f+…)+…((d+e+f+…)
In conclusion stupid video 2/10 two points for having a correct opinion on foil.
1. I respect the utility of FOIL and mnemonics in general.
2. I don't like the table method either. If you watched the video you might have heard me say 'that was nice and easy, we just had to make this giant fucking table', which was my way of saying 'I don't like this method either'.
3. Constants? Most of the video was done with letters which were never specified to be constants or variables.
4. *mnemonic
In conclusion, stupid comment.
I think the better way to deal with polynomial multiplication is thus:
Let N be the number of terms in the first, let M be the number of terms in the second.
Draw a grid of size NxM
Write the terms of the first polynomial along the top, above, of the grid, write the terms of second along the right, beside, the grid.
For every cell in the grid, multiply the terms to the top and to the right outside the grid.
Add up all the terms in the grid
Done.
a....b
ac.bc.c
ad.bd.d
ac+bc+ad+bd
The "table method" was always called the "box method" in my education, and I'm its most arduent defender anytime someone brings up FOIL. It has the added benefit of making the 3blue1brown video on convolutions feel more satisfying.
this is fantastic, but i'm honestly distracted by how consistently thick your sharpies are. do you ever get duds that are thin?? these are all fantastically bold
The duds go in the bad box, and the good ones go in my good box - which are used for the videos!
FOIL was one method that I remember being taught in school. But I never use it because it's pretty confusing in my opinion. One of my math teachers also taught the table method. Which is what I prefer to use, because it's easy to setup and use compared to FOIL. Even the math teacher who taught it prefers this method.
As to mnemonics, my personal wrath of math is aimed at PEMDAS. I think that most middle schoolers would grasp the concept of a field (using Q as an example). Having that, one could replace a division sign b with a *(1 / b) and proceed with the usual rules for multiplying and adding in a field.
Once, long ago, when I was in university, I asked a couple of my non math major friends if they'd ever heard of the distributive law. That led to some teaching moments.
Based on your presentation, I think I could write an inductive proof where I claim to prove (in the spirit of Euclid) that the Foil Method May Be Extended Indefinitely. I can't see any reason why anyone would want to do that.
Interestingly enough, Euclid never said that he attempted to prove that there were infinitely many primes. He said that primes could be produced "indefinitely" which amounts to the same thing, but infinity itself was never on Euclid's menu.
something that my 7th grade math teachers taught was instead of PEMDAS was GEMA standing for: Groupings (parentheses, fractions, so on), Exponents, Multiplication (and division since you can multiply by a reciprocal), and Addition (and subtraction since you can add a negative)
@@lillyflower7834 I think that you've made my point. Thank you!
What’s so interesting about Euclid’s word choice? The statements “there are infinitely many primes” and “primes can be produced indefinitely” are logically equivalent.
@@isaiah0xA455 That's what I said or, at least implied.
I vaguely remember this being mentioned in school, but I barely paid any attention in math class and just did my own thing since I learn best when I see the material and then figure out why that is for myself. So I proved all these things on my own and found my own methods to do things. Teachers didn’t like that I didn’t show my work but I always got the right answer and never cheated so they kinda just had to deal with it.
The only time a use a semi foil method is when multiplying arbitrary polynomials of arbitrary degree in a polynomial ring. It helps group terms with same degree. (Also for formal power series where an infinite table might be cumbersome)
When you learn foil, you're learning about binomial expressions. So you use a method for binomials. When you learn about expressions including more terms, then you can learn another method. The best method is the one that works for whatever specific application you're dealing with. Generality is not inherently desirable.
No one who learns foil is ever going to use it to solve your example with the literals.
That's fair but it leaves many students confused when they encounter things beyond binomials because all they know is the FOIL magic trick; when if they had just learned that to multiply binomials they use the distributive property (which they already know), they wouldn't even need to be taught how to deal with bigger examples, they would just know how to apply properties of mathematics they've mastered before even to new situations.
@WrathofMath Of course not. When they are introduced to a new concept, they're also introduced to a new method for handling the concept. Just like when you learn any new concept in math.
I’ve encountered many students who are confused by concepts more basic than expanding polynomials. When they get FOIL, it gives them a boost of confidence that they need to keep going. There’s plenty of time to teach them each by each or the grid method as they progress.
Matrix products are particular and limited to 2 dimensions, but expression products exist for all finite grids of 1+dimensions.
Foil technically does give you how to multiply any polynomial by any polynomial. Simply bracket the sum inside the polynomial so that there's only one main addition, then foil, remove the brackets, and repeat. It thus technically is all you need for any n-degree polynomial multiplication
for example: (a+b+c)*(d+e) = (a+(b+c))*(d+e) = ad + ae + (b+c)*d + (b+c)*e = ad + ae + bd + cd + be + ce
Edit: lol I commented before watching the full video! Glad this was addressed.
I always like that table method. It works with any kind of multiplication even if you're doing something like converting 245×63 into (200+40+5)(60+3)
Foil is silly yes, but it is a good first method of learning expanding these things out, I was taught distribution to each, then the area method as a easier way to write out long ones, and foil for those smaller expands. Foil is helpful for teaching beginners just a quick method
honestly i would use an area model. SOOOO much less complicated sounding
Mathematicians use FOIL for large numbers, broken into two parts. For instance, (z+a)(z+b)=zz+za+zb+ab=z(z+a+b)+ab
10:10 I use a method inspired by base 10 number column multiplication and omit the x to the power just like omitting 10 to the power in the column multiplication. So treating x as a “base” kinda
I think I tend to treat FOIL as a sort of "canonical ordering" when expending products of two binomials, and then extend that canonical ordering to more complex products, giving one less thing to think about. Said canonical ordering by the way does not match the given "mnemonics" for trinomial and qutranomial products. Rather, it's the first term on the left times everything in the right, plus the second term on the left times everything in the right, etc. Only after expanding each pair of terms do I collect like terms. Which stage I normalise the ordering of different variable factors depends on whether the product is commutative or not.
I was thinking, it might just be easier, assuming distributing a product over a sum is introduced first, to just say "Hey, this left sum is also... just a term itself we can distribute. So we get a sum of these products, then we do normal distribution over each sum term, and just add them all together.
It doesn't introduce any new ideas or require any new mnemonics, it just re-uses old ideas. Then, once students are able to compute expressions like that, maybe introduce some shortcuts to make it easier.
Obviously, FOIL wouldn’t be used for anything but binomials. The broader term would be distribution.
I guess my math textbooks were remiss because I missed this mnemonic in my education. Why do you even need a mnemonic to distribute the terms? Sure, it's easy to miss one, or several, but that's not because you didn't remember what to do, you just did it sloppily. There is no need for a pneumonic here.
Now if there was one for _factoring_ a polynomial, I'd love that (there probably is. I clearly missed the chapter on mnemonics lol)
The moment i learnt about variables, coefficients, terms, and polynomials, i immeadiately knew to multiply two binomials i just use the distributive property. I don't think you should just teach people foil and abstract it that way, you should let them know how the distributive property applies to polynomials and it might help with even multiplying trinomials or bigger polynomials.
"each by each" is also a reasonable definition of multiplication, since 3 x 4 = (111)(1111) = 1111 1111 1111 = 111 111 111 111
FOIL is fine, it's a concrete example of a general concept to be studied later. "Let's reexamine FOIL in a more general way..." can't be uttered until you've learned it
Grateful to have never been taught "first, outside, inside, last" 🙏
Im definitely using the table method from now on
Funnily enough I had never even used the term “foil” used before a week ago, I just always followed the multiplication to its logical conclusion, as I was homeschooled for most of my education and it seems to have, for the most part, been beneficial for my learning, I would never just learn a method to do it without thinking using rule such as “foil” ever like it was a spell, I would much rather truly learn the logic behind it creating a far better intuition for things like math as it serves much better for everything in life. There my essay is done(sry for rambling)
I must have missed the day “foil” was explained because I was literally 27 years old when I learned it was an acronym haha. I just thought it was a weird bit of school math jargon and always felt that “unfoil” would have made more sense.
I’ve always been partial to the “ box method” which seems very similar to your tabular method, except instead of multiplying, just the coefficient you multiply with the coefficient and the given power of X (it actually works for any set of variables as well, such as trying to multiply (ax by) and (cx^2 fy^3) for example). This means you don’t have to include all the tedious zeros in your tubular method, but still allows the handling much larger polynomials. ua-cam.com/users/shortsYOUzwI6wX_s?si=HTofRxiFtpCF4HfM this is a good video demonstration of it.
I dont think ive seen a single person expand (2+3)(4+5) using FOIL before
Is WLOG always just a summoning and invocation of the -demon- axiom of choice?
Underrated video wtf
A generalization of multiplying any nxnomial by any mxnomial is to do a dot product of the two equations. You will always get the correct answer by doing this.
What exactly is a dot product of equations? For that matter, what are the two equations in question? A polynomial is an expression, not an equation.
Don’t you mean a convolution?
Dot product doesn’t seem to make sense here
@@isaiah0xA455 A simple linear equation, the coefficients can be expressed as a vector or matrix. Taking the dot product of the coefficients is the same as what is shown in this video
@@krwada The dot product only multiplies like terms by like terms. It would leave out the “outside, inside” part of the FOIL algorithm. It doesn’t resemble the table multiplication at all.
FOILM(FM)(MF)(ML)(LM) GOT ITS OWN FULL VIDEO!!
I was taught FOIL in 8th and 9th grade. I refused to use it and I just used the box/table method.
You're a man of principle!
The box method is so much better
FOILM(FM)(MF)(ML)(LM)!
So simple!
the Box method wayyy better than foil
Me, who has used the box method for over 12 years:
(I call it box over table)
I like that table method, that would have been handy to know years ago xD
No it wouldn’t have, it is very slow. Just distribute each term of the first polynomial to each term of the second polynomial. This guy is complaining about foil and proposes an even worse method.
I don't like the table method either, and I denigrate it in the video. But my role here is to hate on FOIL, so if someone prefers a different method I fully support that
HAHAHAHAHA. That's one of the reasons I am sibscribed to this channel. This guy is reasonable. Of course FOIL is useless.
Unless you chose to study maths further than you are required to, people who learn FOIL in school are only ever going to need to multiply binomials, not trininomials, and you only need common sense to know not to use it on (2+3)(4+5) because when you are taught FOIL you are taught it for when you CANT combine the terms in the brackets, such as (2x + 3)
If you do study maths as your chosen subject, then you’re probably smart enough to know to disregard FOIL as you move into any polynomial multiplication more complicated than binomials
Weird Al would beg to differ
I love how sincerely you hate FOIL, not even for a meme. And those sarcastic jokes... Brilliant
one must let the hate flow through him
I never, ever heard of FOIL before.
I envy you!
WE'RE FOILS, WE DIDN'T SEE THIS THROUGH!
I think it is not a problem with FOIL but how we teach math.
US is not the only place FOIL is commonly use.
Hey, foil cards are cool, don't- oh, you mean the multiplication thing. Carry on
I think that if you accept any Mnemonics, FOIL isn’t a bad one compared to many others.
If it’s presented carefully, the “FOIL Method” can actually help students remember how to use distributivity of multiplication over addition.
Your concern about using this method to compute products of numbers that are written as sums also is, in my opinion, a bit overblown, especially with that clickbait thumbnail that suggests an incrimination towards those who teach using that particular Mnemonic.
Consider the problem of multiplying two matrix binomials:
(A+B)(C+D)=AC+AD+BC+BD.
Did I use the “FOIL” mnemonic to do that? Yes, but I didn’t write it out. Writing the following would have been wrong, however:
(A+B)(C+D)=CA+AD+CB+BD.
Of course, if you know, you know, that matrix multiplication isn’t commutative. A problem with teaching binomial expansions using the “FOIL” mnemonic, teachers shouldn’t apply commutativity in a first step, only because not all students are going to never take linear algebra or matrix algebra, or some other form of Noncommutative algebra. The kind of example that you exhibited involves numerals that are small in value, so that certain useful mental arithmetic tools aren’t well illustrated, but that’s exactly one I’d the reasons the “FOIL” mnemonic is taught. To review that scenario (with small numbers again but not the same), we see that a more detailed illustration would be the following “proof” that a product of 7 with 8 is 56:
(5+2)(5+3)=(5+2)(5)+(5+2)(3)=(5)(5)+(2)(5)+(5)(3)+(2)(3)=25+10+15+6=35+15+6=56.
Now, if you teach a second grader to do arithmetic this way on certain problems, they learn how the properties of arithmetic work for the step by step justifications. Then during a parent teacher conference you’ll be accosted by parents who already can multiply pairs of one digit numbers whose product odd a two digit result, but their second grader isn’t there yet. Why would you be accosted (verbally)? Well, when the child did their homework, they asked for help from a parent who isn’t equipped to explain the steps or how the distributive law applies in this case. If someone knows how to justify each step above, then they can also justify some useful things about products of larger numbers. For instance, consider the problem of multiplying 435 by 263:
(435)(263)=(400+35)(203+60)
=(400)(203+60)+(35)(203+60)
=(400)(203)+(400)(60)+(35)(203)+(35)(60)
=(4)(10^2)(200+3)+24(10^3)+(35)(200+3)+(30+5)(60)
=(4)(10^2)(200)+(4)(10^2)(3)+24(10^3)+(35)(200)+(35)(3)+(30)(60)+(5)(60)
=(8)(10^2)(10^2)+(12)(10^2)+24(10^3)+(70)(10^2)+(30+5)(3)+(18)(100)+(30)(10)
=24(10^3)+(8)(10^4)+(12)(10^2)+(7)(10^3)+90+15+(18)(10^2)+(3)(10^2)
=8(10^4)+(24+7)(10^3)+(12+18+3)(10^2)+90+15
=11(10^4)+(1+3)(10^3)+3(10^2)+100+5
=114405.
This is by far a not most efficient solution, and one would not use it to solve problems quickly, but to illustrate how the laws of arithmetic work so you can teach yourself some speed for mental arithmetic. Doing no problems in this much detail would make the mental math tricks look more like magic, and so a healthy mixture of methods is better. Here’s a way to use the “FOIL” method on the above problem instead:
(435)(263)=(500-65)(200+63)
=(500)(200)+(500)(63)-(65)(200)-(65)(63)
=(5)(2)(10^4)+(5)(63)(10^2)-(130)(10^2)-(60+5)(70-7)
=(10^5)+(315)(10^2)-(13)(10^3)-((60)(70)+(5)(70)-(60)(7)-35)
=(10^5)+3(10^4)+(5)(10^2)-(13-1)(10^3)-((42)(10^2)+(35)(10)-(42)(10)-35)
=(10^5)+3(10^4)-(13-1)(10^3)-(42-5)(10^2)+(42-35)(10)+35
=(10^5)+3(10^4)-12(10^3)-37(10^2)+(7+3)(10)+5
=(10^5)+(30-12)(10^3)-(37-1)(10^2)+5
=(10^5)+18(10^3)-36(10^2)+5
=(10^5)+(10^4)+(80-36)(10^2)+5
=(10^5)+(10^4)+(50-6)(10^2)+5
=114405.
Needless to say there are much faster ways to compute this, but the “FOIL” Mnemonic can help one do all the above steps in one’s head, rather than writing it all on paper. Shortcuts are needed for students who participate in arithmetic competitions. Not knowing the “FOIL” method can in some cases cost a student precious time in such an enjoyable competition.
Those who object to showing the above steps may in many cases be those who’d rather you just do a calculator problem instead. Of course, some computations are competitions, but some are not.
Check the steps:
(435)(263)-(400+35)(203+60)=0
(400+35)(203+60)-((400)(203+60)+(35)(203+60))=0
(400)(203)+(400)(60)+(35)(203)+(35)(60)-((4)(10^2)(200+3)+24(10^3)+(35)(200+3)+(30+5)(60))=0
=(4)(10^2)(200+3)+24(10^3)+(35)(200+3)+(30+5)(60)-((4)(10^2)(200)+(4)(10^2)(3)+24(10^3)+(35)(200)+(35)(3)+(30)(60)+(5)(60))=0
(4)(10^2)(200)+(4)(10^2)(3)+24(10^3)+(35)(200)+(35)(3)+(30)(60)+(5)(60)-((8)(10^2)(10^2)+(12)(10^2)+24(10^3)+(70)(10^2)+(30+5)(3)+(18)(100)+(30)(10))=0
(8)(10^2)(10^2)+(12)(10^2)+24(10^3)+(70)(10^2)+(30+5)(3)+(18)(100)+(30)(10)-(24(10^3)+(8)(10^4)+(12)(10^2)+(7)(10^3)+90+15+(18)(10^2)+(3)(10^2))=0
24(10^3)+(8)(10^4)+(12)(10^2)+(7)(10^3)+90+15+(18)(10^2)+(3)(10^2)-(8(10^4)+(24+7)(10^3)+(12+18+3)(10^2)+90+15)=0
8(10^4)+(24+7)(10^3)+(12+18+3)(10^2)+90+15-(8(10^4)+(30+1)(10^3)+(30+3)(10^2)+100+5)=0
8(10^4)+(24+7)(10^3)+(12+18+3)(10^2)+90+15-(11(10^4)+(1+3)(10^3)+3(10^2)+100+5)=0
11(10^4)+(1+3)(10^3)+3(10^2)+100+5-114405=0
+++++++*
(500)(200)+(500)(63)-(65)(200)-(65)(63)-((5)(2)(10^4)+(5)(63)(10^2)-(130)(10^2)-(60+5)(70-7))=0
(5)(2)(10^4)+(5)(63)(10^2)-(130)(10^2)-(60+5)(70-7)-((10^5)+(315)(10^2)-(13)(10^3)-((60)(70)+(5)(70)-(60)(7)-35))=0
(10^5)+(315)(10^2)-(13)(10^3)-((60)(70)+(5)(70)-(60)(7)-35)-((10^5)+3(10^4)+(5)(10^2)-(13-1)(10^3)-((42)(10^2)+(35)(10)-(42)(10)-35))=0
(10^5)+3(10^4)+(5)(10^2)-(13-1)(10^3)-((42)(10^2)+(35)(10)-(42)(10)-35)-((10^5)+3(10^4)-(13-1)(10^3)-(42-5)(10^2)+(42-35)(10)+35)=0
(10^5)+3(10^4)-(13-1)(10^3)-(42-5)(10^2)+(42-35)(10)+35-((10^5)+3(10^4)-12(10^3)-37(10^2)+(7+3)(10)+5)=0
(10^5)+3(10^4)-12(10^3)-37(10^2)+(7+3)(10)+5-((10^5)+(30-12)(10^3)-(37-1)(10^2)+5)=0
(10^5)+(30-12)(10^3)-(37-1)(10^2)+5-((10^5)+18(10^3)-36(10^2)+5)=0
(10^5)+18(10^3)-36(10^2)+5-((10^5)+(10^4)+(80-36)(10^2)+5)=0
(10^5)+(10^4)+(80-36)(10^2)+5-((10^5)+(10^4)+(50-6)(10^2)+5)=0
…
1 hours gang?
touch grass, it's a good mnemonic
Over & over & under & under.
Ok don't foil your self
i love ur channel, can i please get a shoutout someday mr. wrath of math?
No
Have you considered the possibility that you might need to GET A LIFE?
average FOILer