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I would like to understand why polynomial multiplication works the way it is defined. For example, we know that $(x+1)^2 = x^2+2x+1$, but how can we prove that this actually works?

More generally, how did we came up with the method used to multiply polynomials?

Any insight would be appreciated.

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What do you mean by "it works"? –  Mariano Suárez-Alvarez Nov 2 '10 at 2:29
There is no "method" to multiply polynomials. Multiplication of numbers or pointwise multiplication of functions is a definition. As crasic mentions, real numbers satisfy a distributivity property and what you're calling a "method" is just the distributivity property manifesting itself in the product of two identical functions "$x+1$". –  Ryan Budney Nov 2 '10 at 2:38
I don't understand the question myself; it's just the straightforward application of the intrinsic properties of addition and multiplication of numbers on polynomials. $x$ is in a sense a "number", except that we do not know its value. –  J. M. Nov 2 '10 at 3:10
I understand that my question is so fundamental that it was hard to understand, but crasic nailed it perfectly. –  Martin Cote Nov 2 '10 at 3:16
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3 Answers

up vote 9 down vote accepted

We define the property that multiplication has to distribute over addition.

That is $A(B+C) = AB + AC$

With polynomials we try to do something similiar, analogous to the real numbers we all know so well.

$(x+1)^2 = (x+1)(x+1) = (x+1)x + (x+1)1 = (x^2 + x) + (x + 1) = x^2 + 2x + 1$

Hence, by the distributive law of multiplication you can see how the defined operation makes sense.

You CAN come up with some binary operation that combines polynomials and doesnt distribute over addition, however this operation would not be called multiplication.

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In addition to crasic's symbolic answer, one can also look at the geometric interpretation of multiplication of two quantities (the area of the rectangle formed with sides of the given length) and find that it is consistent.

$(x+a) * (x+b) = x^2 + ax + bx + ab$

Edit: As pointed out by Douglas, the geometric interpretation is only defined at all when $x$, $a$, and $b$ are positive, so care must be taken to keep algebraic self-consistency when generalizing to the real numbers. Geometric "proofs" should only be used as an "intuition check", in any case.

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Although, we need to be a bit careful when some combination of a,b and x is negative. –  Douglas S. Stones Nov 2 '10 at 3:34
@Douglas It's definitely not a rigorous proof (I said, "in addition to"), and perhaps it would have been more proper to post it as a comment instead of its own answer, but in this, you can sort of grasp the basic idea of polynomial multiplication with geometric interpretations, and take it further with care to keep things self-consistent. –  Justin L. Nov 2 '10 at 4:48
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If you have a set $X$ and two real valued functions $f$, $g:X\to\mathbb R$, then there is a standard way to multiply $f$ and $g$: the product is the function $h:X\to\mathbb R$ such that $$h(p)=f(p)g(p)$$ for all elements $p$ of $X$.

If $X=\mathbb R$ and if $f$ and $g$ are polynomials, then the function $h$ turns out also to be a polynomial, given by the usual formula. This completely determines the multiplication rule.

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@Mariano: So... polynomials are identical with polynomial functions? (Yes, they are over $\mathbb{R}$, and the distinction is likely to be confusing to Mr Cole, but still...) –  Arturo Magidin Nov 2 '10 at 2:37
For the current purposes, yes :) –  Mariano Suárez-Alvarez Nov 2 '10 at 2:39
@Arturo, I suspect the idea of an abstract polynomial independent of functions is a foreign idea to anyone who hasn't studied fairly abstract mathematics. –  Ryan Budney Nov 2 '10 at 2:40
@Ryan: I suspect that I don't suspect that, I know it. (-: Not sure if we should be encouraging it, though, when it is not exactly required for this... –  Arturo Magidin Nov 2 '10 at 2:45
@Ryan: Firstly, I wasn't particularly talking about polynomial multiplication, but polynomials in general. I would say that polynomials are like Lego instructions. If you know how to add, scale, and multiply then polynomials tell you how to construct new things out of old things. Simple! –  Andrew Stacey Nov 2 '10 at 20:32
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