HP 48gII - Manual (Page 189)

Page 5-18
Note: Refer to the help facility in the calculator for description and examples
on other modular arithmetic. Many of these functions are applicable to
polynomials. For information on modular arithmetic with polynomials please
refer to a textbook on number theory.
Polynomials
Polynomials are algebraic expressions consisting of one or more terms
containing decreasing powers of a given variable. For example,
‘X^3+2*X^2-3*X+2’ is a third-order polynomial in X, while ‘SIN(X)^2-2’ is a
second-order polynomial in SIN(X). A listing of polynomial-related functions
in the ARITHMETIC menu was presented earlier. Some general definitions on
polynomials are provided next. In these definitions A(X), B(X), C(X), P(X),
Q(X), U(X), V(X), etc., are polynomials.
• Polynomial fraction: a fraction whose numerator and denominator are
polynomials, say, C(X) = A(X)/B(X)
• Roots, or zeros, of a polynomial: values of X for which P(X) = 0
• Poles of a fraction: roots of the denominator
• Multiplicity of roots or poles: the number of times a root shows up, e.g.,
P(X) = (X+1) 2 (X-3) has roots {-1, 3} with multiplicities {2,1}
• Cyclotomic polynomial (P
n
(X)): a polynomial of order EULER(n) whose
roots are the primitive n-th roots of unity, e.g., P
2
(X) = X+1, P
4
(X) = X 2 +1
• Bézout’s polynomial equation: A(X) U(X) + B(X)V(X) = C(X)
Specific examples of polynomial applications are provided next.
Modular arithmetic with polynomials
The same way that we defined a finite-arithmetic ring for numbers in a
previous section, we can define a finite-arithmetic ring for polynomials with a
given polynomial as modulo. For example, we can write a certain polynomial
P(X) as P(X) = X (mod X
2 ), or another polynomial Q(X) = X + 1 (mod X-2).
A polynomial, P(X) belongs to a finite arithmetic ring of polynomial modulus
M(X), if there exists a third polynomial Q(X), such that (P(X) – Q(X)) is a
multiple of M(X). We then would write: P(X)
≡ Q(X) (mod M(X)). The later
expression is interpreted as “P(X) is congruent to Q(X), modulo M(X)”.