Don't Foil With Me!

Added: 2026-05-11

[00:00:03]Good afternoon all my subscribers and students. I'm making this video because well, I have even more good news. We have now over 100 people have subscribed to the channel in the just in the past 24 to 48 hours. So once again, I want to thank my subscribers. I will be as you can see, I told you in the previous videos, I'm going to up the ante. I really am going to be posting a lot of content and we'll be working very hard for you because that's one of the reasons why I do this. For those of you who don't know me, my name is Alfred Cromwell. I am the founder and president of City Tutoring, a nationwide agency that is going to be dedicated to tutoring, fostering math, and fostering academic excellence across the country and hopefully the world, but we start out humble, of course.

[00:00:52]We should we should always maintain our humility, but I'm saying we always have to be realistic when we first start a new project. So that being said, I wanted to take a moment to describe something that has been bothering me.

[00:01:06]Sorry to say, I mean I'm it's been it bothers me.

[00:01:09]And that is the I keep hearing over and over again there are hundreds of tricks, so-called tricks that I hear from students that they're taught in the schools.

[00:01:20]One of them, the one that we're going to be discussing today is foil.

[00:01:26]Foil.

[00:01:27]Which is taught by I'm not saying all the teachers teach this. I'm not saying all the schools do this, but it's it's taught pretty often enough to make it unacceptable to me.

[00:01:44]And unfortunately it really has become a problem because a lot of students are relying on these tricks and then by the time they get to the college level, they they start running into serious trouble.

[00:01:56]Right? So, FOIL is the lazy person's shortcut.

[00:02:01]It's a nifty little acronym that some high school math teachers everywhere cling to like it's some golden key to algebraic enlightenment.

[00:02:10]It stands It stands for first, outer, inner, last. Simple, right? Yeah, too simple. In fact, it's the perfect tool for those who want to pretend that they're teaching students real mathematics, pure mathematics, without actually doing so. You know, I always see FOIL as the I guess I it would be like the mathematical equivalent of teaching someone to cook by giving by giving them a microwave.

[00:02:33]Right? It gets the job done, I guess, but God forbid they they ever want to understand what's actually happening in the kitchen.

[00:02:41]So, let's be clear here. FOIL only works for multiplying two binomials. That's it.

[00:02:49]So, yeah, it's a trick. It's not a tool.

[00:02:51]We're not here to do tricks, we're here to truly understand mathematics and make mathematics understandable to you.

[00:02:58]If you're relying on tricks, you're not really understanding the procedure.

[00:03:02]It's the You know, FOIL it's it's since it's a trick, it's kind of like the fast food of algebraic techniques. It's it's quick, it might be satisfying, addictive in the moment, but it's really devoid of any real substance or long-term benefits.

[00:03:15]So, I consider algebra to be I'm an algebraist. I'm a pure mathematician.

[00:03:19]I'm an algebraist. It is the art of problem-solving.

[00:03:22]And if that if that is my premise, then FOIL is kind of like the paint-by-numbers version of it. It's shallow, it's color by coding, and it makes students feel like they've mastered something when in reality you've learned nothing at all.

[00:03:36]So, don't you FOIL with me.

[00:03:39]Right?

[00:03:40]So, I'll give you a a classic example that they use, and I put the board up just because otherwise it takes so long to, and I know you like short videos, I know.

[00:03:48]So, let's say that you have the classical binomial multiplication, right? Let's say you had x + 3 parentheses x - 2.

[00:04:01]By now you should know that when two parentheses are together, that means multiplication.

[00:04:07]Now, if you learn foil, it teaches you to multiply the first terms, the outer terms, the inner terms, and then the last terms, slapping them together like some Frankenstein's monster of an expression.

[00:04:20]And you end up with the following. You do x and x, right? Which is of course x squared.

[00:04:26]Again, I'm assuming for this video that you've done this before. If you haven't, then you're not going to really understand the context here. Then the outer is two and x and three x, right?

[00:04:36]So, it's going to be -2x +3x.

[00:04:43]And then what we do is -2 * 3 or 3 * -2 is -6.

[00:04:52]And then they teach you to, of course, combine the like terms. So, you have x squared, -2 + 3 is 1x or just x.

[00:05:00]x squared + x -6. And you would get that um a tri- you would get a trinomial. Congratulations.

[00:05:10]You've learned you you What have you learned though? You've learned how to follow a list of instructions. Would you like a cookie?

[00:05:19]But now, let's try multiplying something a little more complicated.

[00:05:24]How about if I gave you a binomial times a trinomial?

[00:05:29]x + 3 We'll erase all this.

[00:05:41]We're going to multiply that by the trinomial. Let's do, I don't know, hmm, x squared - x plus five.

[00:05:56]Oh, no.

[00:05:58]I dare you to use foil on this.

[00:06:01]Right? What's going to happen with foil now? It just threw up its hands, right?

[00:06:05]It's walked out the door now.

[00:06:07]Foil is utterly useless here.

[00:06:11]You have to just brace yourself actually to solve this. And guess what? Yeah, you're going to have to learn the distributive property. Something you should have been taught from the start.

[00:06:21]Right? Now, if you don't know the distributive property, the basic basic distributive property is that any number outside of parentheses, in this case let's say three numbers, three different numbers, can be distributed literally, multiplied to every element in the parentheses, right? So, a times b becomes ab.

[00:06:40]a times c becomes a plus ac rather. So, ab plus ac. A is of course a factor of both of these terms.

[00:06:50]Right? That's the real way of learning this. The distributive property, it states the the the the definition that you must multiply each term in one expression by every term in the other.

[00:07:03]It's a process that's rooted in logic.

[00:07:05]There's a proof of it. It's it's it's rooted in understanding. And guess what?

[00:07:10]It works for all polynomial multiplications, not just binomials. You want to multiply trinomial by a binomial? No problem. So, you have a cubic expression you want by a quartic expression? Step right up.

[00:07:24]The distributive property does not discriminate. Mathematicians, we do not like ambiguity. Right? And if you don't have the definitions correctly, you're wasting time.

[00:07:34]Right? So, I don't want to hear this whole foil foil foil. I hear it all the time.

[00:07:39]Bothers me. It should bother you by the way because it's going to make you run into trouble if you're not if you at least, you know, when you start running into this and you're not sure to distinguish. Now, some of your teachers might have told you, I'm not saying they didn't. Some of your teachers might have specified that well, it only works for binomials. But why do that?

[00:07:58]You know, it it just doesn't make sense to me. And the distributive property is versatile, it's universal, everything FOIL is not.

[00:08:08]So, let's revisit this expression.

[00:08:11]If we use the distributive property here, or as I like to call it real math, real algebra, you simply multiply each term in the first polynomial, which is the first polynomial is x + 3, by every term in the second.

[00:08:26]Right? So, we now take like so.

[00:08:30]So, we're going to go back to So, this right here, this first term, now you have x and then you multiply by all the terms.

[00:08:38]x squared minus x plus five.

[00:08:47]What else have we got here? Um we also have the the the plus three, right? So, plus three times every element in the second parentheses.

[00:09:04]Literally, this is how it works. And if you understand this, then life is a lot easier.

[00:09:09]Right? And then now you just if you know your rules of exponents, you multiply, I'm going to use white now, x times x squared, that gives you x to the third.

[00:09:19]x times negative x gives you negative x to the square.

[00:09:24]x times five, that gives you plus five x.

[00:09:28]Plus and then you do it for the second one, right? Three x squared because and then three times negative x is negative three x, so you have 3x squared minus 3x plus, right?

[00:09:45]3 * 5, which is 15.

[00:09:51]And then from here you just Well, let's combine them.

[00:09:55]x cubed does not match with anything, so we keep it.

[00:10:00]We cannot combine it with all the like terms. Uh negative x squared plus 3 x squared, that's going to give you 2 x squared.

[00:10:11]5x minus 3x, that's 2x plus 15.

[00:10:20]No clever mnemonic demonic I was going to say demonic, but it's not that big of a deal, but no clever devices needed, just pure unadulterated mathematical reasoning.

[00:10:33]And here's the best part. You can apply this method to any polynomial multiplication, no matter how large.

[00:10:40]Now, FOIL meanwhile sitting in the corner whimpering, quaking at the sight of anything with more than two terms.

[00:10:47]The real tragedy here, folks, is Excuse me, is that it encourages laziness and short-term thinking.

[00:10:55]And students don't learn the why of polynomial multiplication, only the how.

[00:11:00]You don't see the beauty of distributive reasoning, the way it mirrors uh the logic really of multiplication and addition. Instead, you memorize, some of you, a trick that works only in one narrow situation, and it leaves you floundering when the algebra gets real.

[00:11:18]And it's like giving someone a bicycle with training wheels, and then expecting them to ride a motorcycle. That's ridiculous. It's preposterous.

[00:11:26]So, FOIL is the symptom of a larger problem in math education. We teach students in this country to pass tests, not to understand concepts. We give them shortcuts instead of teaching them the long road to real comprehension. And the worst part of all this, it's all done in the name of simplicity. I've heard it over and over and over again because apparently to some to some educational institutions, you're too dim to handle the concepts. It's so daunting. It's so hard, isn't it?

[00:11:56]Distributing multiplication, how patronizing are these people? Who are these people, really?

[00:12:02]So, in conclusion that from my video today, FOIL is a crutch.

[00:12:08]It's shiny, perhaps attractive to some who want to take the easy way out in a binomial, but it's an utterly useless crutch. And math teachers, I want to I want math teachers, if you're a math teacher and you're watching this video, I want you to throw that away.

[00:12:23]Don't call yourself a math teacher and FOIL your with your students. Don't FOIL with me, that's for sure.

[00:12:28]It doesn't teach your students how to think.

[00:12:31]It teaches students how to jump through hoops.

[00:12:33]And while it might earn them a decent grade on one of your quizzes, it will leave them stranded when they reach more advanced mathematics. I've had countless number of students, the first year they're in my class, they start crying.

[00:12:46]Not because of me necessarily, well some because of me, but some of them will tell me, "I've never failed a test in my life." Really? But you can't do basic algebra.

[00:12:55]Yeah, but your teachers only gave you a passing grade, right? They didn't want to make you feel bad.

[00:13:00]That's not how we operate here at City Tutoring. We make sure that you actually do understand. And if we have to start at the beginning, we will, but it's all to your benefit. Do yourself a favor.

[00:13:12]Stop using tricks. Start truly understanding mathematics.

[00:13:17]Right? If it makes it be slower for you, that's okay. There's nothing wrong with being slow.

[00:13:23]Right? There's nothing wrong with doing something a slow way as long as you're truly understanding.

[00:13:27]Uh let's aim higher. Let's aim higher, folks.

[00:13:31]Real math is about depth, not tricks.

[00:13:36]Are we I mean, are we satisfied to let the next generation think that multiplying binomials is the pinnacle of some kind of a achieve You haven't achieved anything with that.

[00:13:46]So, I think FOIL has done enough damage.

[00:13:50]There are other tricks that we'll be talking about. Um We'll be talking about the how everything just sort of um There's another trick that they use Well, not trick, but they use PEMDAS.

[00:14:04]Don't you dare.

[00:14:05]Don't you dare.

[00:14:07]We're going to talk about PEMDAS as well.

[00:14:09]So, stay tuned as always. Um I'm probably probably this week this will be the last video this week. I'll try to I'll put another one this week, but I want to be able to solve module two. I haven't looked at module two of the SAT.

[00:14:23]I will be looking at it potentially this week and or maybe even Friday. We'll see. I'll post a video on that. Let's see if it's so hard as people say it is.

[00:14:29]People say They keep telling me They keep telling me module two of the SAT is so difficult. It's complicated. The timing. Well, let's see if that's true.

[00:14:39]If this was helpful to you, if you agree with me, if you think that these kind of tricks are misleading students and you know, leading to a misunderstanding of things, please let me know. Um and please subscribe to my channel. Leave a comment down below.

[00:14:53]And thanks to you Thank you once again for watching.