Picture proof for expansion of
Most students fail to intuit
After substituting
I couldn't find a picture proof from Roger B. Nelson's Proofs without Words (1993),
Proofs without Words II (2000), or
Proofs without Words III (2015).
2 answers
Here's a semi-visual, semi-algebraic approach.
A useful alternative perspective on (univariate) polynomials is to think of them as being represented by their sequence of coefficients – filling in zeroes for any missing monomials. With this perspective, multiplying by
0 | 1 | 2 | 3 | Corresponding polynomial | |
---|---|---|---|---|---|
(1) | |||||
(2) = |
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(3) = |
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(4) = (2) + (3) |
This is, of course, the typical algebraic proof just written with a different notation for polynomials, but I think it makes it pretty obvious how to arrive at the solution and why it is correct and will generalize. I do find viewing polynomials as their sequences of coefficients does often make what is happening clearer. Theoretically, a benefit of defining operations in terms of this sequence perspective, is that you are always operating on a normal form representation of the polynomials, so you never end up with the same polynomial presented in different ways.
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