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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"?2010-11-02
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    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 "$x+1$".2010-11-02
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    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.2010-11-02
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    I understand that my question is so fundamental that it was hard to understand, but crasic nailed it perfectly.2010-11-02

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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.2010-11-02
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    @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.2010-11-02
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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...)2010-11-02
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    For the current purposes, yes :)2010-11-02
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    @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.2010-11-02
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    @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...2010-11-02
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    @Ryan: that's a disheartening point of view! I think that the idea of abstract polynomials is actually quite intuitive, if looked at in the right way, and very accessible at quite a low-level.2010-11-02
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    @Andrew: In what sense is it low-level, I mean, what is your access-point for the concept, why would you define polynomial multiplication the way its defined if you do not have the notion of function or a fairly abstract (say, functorial) reasoning to support it?2010-11-02
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    @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!2010-11-02