The pcoef function, The proot function, The ptayl function – HP 50g Graphing Calculator User Manual

Page 198: The quot and remainder functions, The pcoef function ,5-21

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Page 5-21

The PCOEF function

Given an array containing the roots of a polynomial, the function PCOEF
generates an array containing the coefficients of the corresponding polynomial.
The coefficients correspond to decreasing order of the independent variable.
For example: PCOEF([-2,–1,0,1,1,2]) = [1. –1. –5. 5. 4. –4. 0.], which
represents the polynomial X

6

-X

5

-5X

4

+5X

3

+4X

2

-4X.

The PROOT function

Given an array containing the coefficients of a polynomial, in decreasing order,
the function PROOT provides the roots of the polynomial. Example, from
X

2

+5X-6 =0, PROOT([1, –5, 6]) = [2. 3.].

The PTAYL function

Given a polynomial P(X) and a number a, the function PTAYL is used to obtain
an expression Q(X-a) = P(X), i.e., to develop a polynomial in powers of (X- a).
This is also known as a Taylor polynomial, from which the name of the function,
Polynomial & TAYLor, follow:

For example, PTAYL(‘X^3-2*X+2’,2) = ‘X^3+6*X^2+10*X+6’.

In actuality, you should interpret this result to mean
‘(X-2) ^3+6*(X-2) ^2+10*(X-2) +6’.

Let’s check by using the substitution: ‘X = x – 2’. We recover the original
polynomial, but in terms of lower-case x rather than upper-case x.

The QUOT and REMAINDER functions

The functions QUOT and REMAINDER provide, respectively, the quotient Q(X)
and the remainder R(X), resulting from dividing two polynomials, P

1

(X) and

P

2

(X). In other words, they provide the values of Q(X) and R(X) from P

1

(X)/P

2

(X)

= Q(X) + R(X)/P

2

(X). For example,

QUOT(X^3-2*X+2, X-1) = X^2+X-1

REMAINDER(X^3-2*X+2, X-1) = 1.

Thus, we can write: (X

3

-2X+2)/(X-1) = X

2

+X-1 + 1/(X-1).

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