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**23. Polynomials**

The monadic function` M=: 3: * ] ^ 2: `
is a multiple of an integral power of its argument,
and is called a *monomial*; and a sum of monomials
such as` SM=: (3:*]^2:)+(2.5"_*]^4:)+(_5:*]^0:) `
is a *poly*nomial.

Any polynomial can be expressed in the *standard*
form` c&p`,` `where` c `is a suitable list
of *coefficients*, and where` p=: +/@([*]^i.@#@[)"1 0` .
For example:

SM=: (3:*]^2:)+(2.5"_*]^4:)+(_5:*]^0:)
p=: +/@([*]^i.@#@[)"1 0
c=: _5 0 3 0 2.5
x=: _2 _1 0 1 2
(SM x),(c p x),:(c&p x)
47 0.5 _5 0.5 47
47 0.5 _5 0.5 47
47 0.5 _5 0.5 47

The primitive` p. `is equivalent to the
function` p `defined above, and will be used hereafter.
The polynomial` c&p. `is very important for a
number of reasons, including:

1. It applies to any numeric argument, real or complex
(and the parameter` c `may also be complex).

2. It can be used to approximate a wide range of functions.

3. It is *closed* under a number of operations;
that is, the sum, difference, product, the composition` @`,` `
the derivative, and the integral of polynomials
are themselves polynomials.

4. The coefficients of the results of each case listed in
3 are easily expressed. For example, if` #c `
equals` #d`,` `then` c&p. + d&p. `
is equal to` (c+d)&p.` . More generally,
it is equal to` (+/c,:d)&p.` . Thus:

ps=: +/@,: Polynomial sum
pd=: -/@,: Polynomial difference
pp=: +//.@(*/) Polynomial product
D=: d.1 Scalar (rank 0) first derivative
pD=: 1: }. ] * i.@# Polynomial derivative
pI=: 0: , ] % 1: + i.@# Polynomial integral

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