monomials

You don't have to talk about integrating higher degree polynomials, because a polynomial is just a sum of monomials, and you just end up integrating monomials over and over again.

A very rough intuition of the idea is that a polynomial p(z) doesn't vanish if one of its monomials is significantly larger than all the other.

A log-log graph is useful for turning monomials into lines, which is exactly what a Bode plot is.

>>> monomials[0](2.0), monomials[1](2.0), monomials[2](2.0) (8.0, 8.0, 8.0) The i variable was captured as an address which all the lambdas share.

Another application of arbitrary-length sequences is storing sparse multivariable polynomials as maps from monomials to coefficients.

Algebraic closure can be interpeted as saying that any rational functions can be factorised into monomials.

But the number of different possible monomials in a polynomial with bounded length is obviously exponential.

The equation defining this curve is degree 3 but inhomogeneous, meaning that not all monomials have the same total degree.

Reciprocally, the polynomial has more chances to vanish for values z such that two monomials have the same order of magnitude.

In log-log plots[1], functions of form f(x) = ax^k (monomials) appear as straight lines.

Precisely, you're not repeating p(x) q(x)-times, you've used that p(x) is a linear combination of monomials and then the distributive property of Z[x].

A polynomial's degree is the highest degree of any of it's monomials.

It depends on the "serial numbers of [...] monomials, in the natural order of monomials".

There's nothing "rectangular" about this any more than a polynomial (as a sum of monomials) is "rectangular".

What's particularly difficult is that, confronted with the fact that 25 and 49 are manifestly different numbers, many still cling to the rule that a^2 + b^2 = (a+b)^2, because of the "law of distribution", which they haven't learned as the "law of distribution of multiplication of monomials over addition, and only that".