If we have a finite sequence *a*_{0}, *a*_{1}, . . . , a_{n} of real numbers (written briefly as * a*), then we can look at the polynomial

P_{a}(*t*)= a_{0} + a_{1}*t* + ··· + *a*_{n}t^{n}.

Conversely, given a polynomial *Q* of degree *m ≤ n*, we can recover a unique sequence *b*_{0}, *b*_{1}, . . . , *b*_{n} such that

*Q* (*t*) = *b*_{0} + *b*_{1}t + · · · + *b*_{n}*t*^{n}

by, for example, taking *b*_{k} = *Q* ^{(k)} (0) / *k*!.

We observe that if *a*_{0}, *a*_{1}, . . . ,*a*_{n} , and *b*_{0}, *b*_{1}, . . . , *b*_{n} are finite sequences, then

*P*_{a}(t)*P*_{b}(*t*) = *p*_{a*b}(*t*),

where ** a * b** =

*c*_{k} = a_{0}b_{k} + *a*_{l}*b*_{k-l} + . . . + *a*_{k}*b*_{0},

where we interpret *a _{i},* and b

To see the kind of use that one can make of this observation, ...

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