Polynomials; The Horner Function - HP 48gII User Manual

Graphing calculator.

FACTORS:
SIMP2:
The functions associated with the ARITHMETIC submenus: INTEGER,
POLYNOMIAL, MODULO, and PERMUTATION, are the following:
Additional information on applications of the ARITHMETIC menu functions are
presented in Chapter 5 in the calculator's user's guide.

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).
Functions COLLECT and EXPAND can
be used on polynomials, as shown earlier. Other applications of polynomial
functions are presented next:

The HORNER function

The function HORNER produces the Horner division, or synthetic division, of a
polynomial P(X) by the factor (X-a), i.e., HORNER(P(X),a) = {Q(X), a, P(a)},
where P(X) = Q(X)(X-a)+P(a). For example,
HORNER('X^3+2*X^2-3*X+1',2) = {'X^2+4*X+5, 2, 11}
3
2
2
i.e., X
+2X
-3X+1 = (X
+4X+5)(X-2)+11. Also,
HORNER('X^6-1',-5)=
{'X^5-5*X^4+25*X^3125*X^2+625*X-3125',-5, 15624}
6
5
4
3
2
i.e.,
X
-1 = (X
-5*X
+25X
-125X
+625X-3125)(X+5)+15624.
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