I’ve been looking at this website and I’m puzzled by it. Specifically this bit:
More specifically where that six comes from, which seems like quite a crucial part of it, being where the answer comes from. What I don’t understand is why (3 + Dx)^2 is equivalent to (9 + 6Dx + Dx^2) instead of (9 + Dx^2).
I’ve tried google, but it isn’t helpful.
It’s not even calculus, just basic algebra. Here it is step by step:
(3+x)^2
(3+x)(3+x)
3(3+x) + x(3+x)
9 + 3x + x3 + x^2
9 + 6x + x^2
The above link discusses this situation.
FOIL -> First Outer Inner Last
That is, in order to multiply two binomials you multiply the first times the first, then the outer by the outer, then the inner by the inner, then the last by the last, and then sum the products.
y=(4+3)(5+7)
=(45)+(47)+(35)+(37)
=20+28+15+21
=84
or
y=(4+3)(5+7)
=(7)(12)
=84
It works on variables as well as constants, so
y=(x+2)(2x+4)
=(x2x)+(x4)+(22x)+(24)
=2x^2+4x+4x+8
=2x^2+8x+8
for your equation, you had some Δx terms, but you can just rewrite them as u and then sub back in if they throw you off.
And a picture to help one visualize what’s going on when two binomials are multiplied.
That’s a great picture. I never thought about trying to visualize this. I’m going to print this out for my son, who’s taking algebra this year.
I took Algebra I in the 8th grade. The teacher wrote this on the board and asked who knows the answer:
(x + 2)[sup]2[/sup]
Andy, always quick to answer, raised his hand immediately and said, “X squared plus 4!”
The teacher went into a faux rampage yelling, “A binomial times a binomial is a trinomial!” He repeated it over and over, and jumped on a desk, and started walking around the room on top of the desks, yelling, “A binomial times a binomial is a trinomial!”
That is one of the few moments I have never forgotten from school.
Meantime, Andy went on to become an executive at a video game company and retired in his 40’s on stock options.
I hope you actually are misremembering it, as it isn’t true… a binomial times a binomial leads to four terms generally, not three. (Or did I just get whooshed?)
(a+b)(c+d) = ac + ad + dc + db <---- four terms
In algebra class, the binomials often share a term, with simple numerical values for the other terms, leading to a situation where you can combine two of the four terms in a natural way:
(a+5)(a+6) = a[sup]2[/sup] + 6a + 5a + 30 = a[sup]2[/sup] + 11a + 30 <---- four terms combined into three
Maybe the exact quote was “A binomial squared is a trinomial?”
At any rate, if a rampage, faux or otherwise, would keep students from making the “freshman’s dream” error, I’m tempted to try it.
I don’t think it would. I mean sure, that phrase might help for that one particular mistake but in a general sense, the underlying process of thinking about math is broken.
The problem is that typical kids get through algebra by internalizing step-by-step manipulations as if it was some kind of meta-language. You follow this rule, memorize these steps, etc to get the “correct” answer. You can certainly get A grades this way. It’s just a wild guess but maybe less than 1% of kids finishing algebra realize that it’s actually just a generalization of concrete numbers. If they truly internalized that, they’d easily see the binomial mistake without memorizing English rules:
=(2 + 3)^2
=(2^2 + 3^2)^2
=(4 + 9)
=13 <– hmmm is that right???
What if I substitute by adding the numbers inside the parens first…
=(5)^2
=25 <– why do I get 25 this way but 13 the other way?
What kind of answer does FOIL method give me…
=2^2 + 23 + 32 + 3^2
=4 + 6 + 6 + 9
=25 <– now I see why FOIL works the way it does, that 13 answer must be bogus
It’s similar to derivatives for trig:
d/dx sin x = cos x
d/dx cos x = sin x <– (or is it -sin x ??? I can’t remember… why is there minus sign in front of sin, that makes no sense!!!)
Typical (struggling) students are trying to see some kind of consistent meta language (as if they expect math to be consistent and show symmetry at the English grammar level.) The crazy part as you can actually get passing grades with this defective thought process. However, the serious student can derive all the trig derivatives by hand from the unit circle and see that yes, it’s actually “- sin x” even though it looks “weird” written in English.
I don’t blame math teachers for this because they don’t really have the time to ensure that students are actually doing and understanding real math instead of pseudo-math. It would take a lot of work to continually ask all year, “Johnny, I see you’ve made a mistake here; are you doing real math or trying to memorize some meta-English-syntax math?”
not to mention (x-1)(x+1) = x^2-1, a binomial
I appreciate the flaw in the assertion. I am not misremembering, unless, of course, I am. That is to say, the memory is extraordinarily vivid, but I know memories are only worth the neurons they are printed on.
Oddly enough I met that teacher by happenstance some 20 years later. He was subcontracting to the company I worked for.
I’m a big fan of visualization. I’m always trying to “see” what I’m doing. For example, if I need to multiply a couple of two-digit numbers I break them down into their tens and ones constituents and then do the four simple multiplications. That picture keeps me from forgetting one. If you liked that, then you might also like this. The Pythagorean Theorem animated. And another one.
Well, technically, it is still 2nd order polynomial of the form ax[sup]2[/sup]+bx+c, its just that b=0 in this case. This is a good example of where an algebra student would memorize a special solution, without understanding whats really going on.
Really? The only way I was taught that was to realize that it was going to be (x+1)(x-1) = x[sup]2[/sup]+x-x-1 = x[sup]2[/sup]-1. Are there really math classes that would just teach it as some sort of rule, without proving it first?
Technically it is both a 2nd order polynomial of the form ax[sup]2[/sup]+bx+c and a binomial.
This is a good example of someone being wrong.
A binomial is simply something with 2 terms. X^2-1 has two terms.
See here the example of a monomial Monomial Definition (Illustrated Mathematics Dictionary)
3x^2 is a monomial, even though it could be written as 3x^2+0x+0.
And the example of the trinomial has a y^2 term, but no y term.
The difference of two squares can be factored out as the square root of the first term minus the square root of the second term times the square root of the first term plus the square root of the second term.
I have no idea if I was taught to solve it that way or not, algebra was a long time ago.
