Well, do it. It is multiplication. (x-5)*(x-5) Go!
...then forget you EVER heard of "F.O.I.L". Simply learn how to multiply. Have you ever done it with pencil and paper? Many never do, these days.
FOIL doesn't work with ANYTHING but a pair of binomials.
(a+b+c)(d+e) -- Use the "111221223132" method!! Utterly silly. Do NOT memorize a method that has application to only one circumstance.
(a+b+c)(d+e+f) -- Use the Convolution Method. What? (There is such a method, but it is not applicable, here.)
Really, just learn to multiply? Each term from one factor matched with each term from the other factor. That's what there is to it.
2) You can use FOIL on (a + b + c)(d + e) = ((a + b) + c)(d + e).
d) (reference to pop-culture intended)
p.s. The binomial theorem ONLY works on binomials...
Why, oh why, do we always get to this "offended" thing? Who was offended? I dare you to point out where offense was expressed. Strong opinions, honesty, a few exclamation points - these do not have to be offensive. All are just as free to take them at face value as they are free to be offended. Feel free to pick the former once in a while.
Forum. Discussion. Views and opinions. It's rather what we do, here, and on most other public forums.
There is never anything so obvious that every student will be benefitted by it. Many students need a little encouragement to think about what it is they are doing, rather than simply memorizing tricks.
I learned "FOIL" back in my dark ages, too. Then, I got over it and rememberd how to multiply. I am convinced that "FOIL" has confused, compartmentalized, and limited the success of far more students than it has helped or freed to learn how to think. It should be removed from textbooks except as an historical note concerning how we quite deliberately used to limit students' success and contribute to math anxiety.
My views. I welcome others'.
p.s. Hardly a valid comparison. "FOIL" addresses only multiplication of two binomials. The binomial theorem has myriad applciations.
The last step can be most easily seen by noting that it is the reverse of a factoring problem: Factor the x - 5 from both terms:
Okay, so now multiply everything out and you will get the same result as FOIL.
Why is this important? Because, to use an example that TKHunny mentioned (a + b + c)(d + e) cannot be done using FOIL. But if you break each factor down:
(a + b + c)(d + e) = a(d + e) + b(d + e) + c(d + e)
the problem becomes fairly simple, if a bit long at your level.
(a + b + c)(d + e) =
((a + b) + c)(d + e) =
First: (a + b)(d)
Outer: (a + b)(e)
In some first semester of Abstract Algebra somewhere in the known universe, I'm certain that polynomial multiplication is an exercise in Induction on the distributive axiom.
I don't suggest that students learn Pascal's Pyramid (for expanding powers of trinomials), but rather that they employ a similar technique and stick with the Triangle/Binomial Theorem.
Yeah... I need more coffee too!
2) I did forget to say one thing. While I firmly believe that no single method will benefit all students, we also should hear the other side. Hardly anything would be of value to no one. Since you marked "thanks" almost immediately on the original "FOIL" post, the rest was just for discussion and perhaps, if you were up to it, to broaden your horizons.
Well, broadened or not, we're all delighted we could help, even if that very first "FOIL" post was of the most help to you, personally, in this particular case. It's students who never come back and we never know if they saw the response that might generate a bit of annoyance. My first reply was just to get you to show us something besides your ability to type in the question. Don't forget to come back if you get stumped again. And please show some work. :-) Don't worry. I don't have a soap box for everything.
You'all are totally cracking me up. It's a little like a Ceasar Salad without the anchovies (whether actual fishies or just in the Worchestershire sauce). In other words, NOT a Ceasar Salad.Originally Posted by TheChaz
I have to agree with you entirely. Many, many things can be done with "FOIL", if you do something else and then call it "FOIL". "FOIL" is fundamentally a beginning mnemonic device for the multiplication of two simple binomials, and it has not ever been anything else. Don't try to elavate it to Sylow or Galois.
Throwing names around doesn't prove any point (except maybe that you've glanced at the latter half of an algebra book ).
There is no mention of trinomials in rings axioms; therefore we invoke the distributive property multiple times to justify the multiplication of trinomials.