HP 15c - Manual (Page 13)

Section 1: Using _ Effectively 13
This function equals zero at no more than n real values of x, called zeros of the polynomial.
A limit to the number of positive zeros of this function can be determined by counting the
number of times the signs of the coefficients change as you scan the polynomial from left to
right. Similarly, a limit to the number of negative zeros can be determined by scanning a new
function obtained by substituting −x in place of x in the original polynomial. If the actual
number of real positive or negative zeros is less than its limit, it will differ by an even
number. (These relationships are known as Descartes' Rule of Signs.)
As an example, consider the third-degree polynomial function
f(x) = x
3
− 3x
2
− 6x + 8 .
It can have no more than three real zeros. It has at most two positive real zeros (observe the
sign changes from the first to second and third to fourth terms) and at most one negative real
zero (obtained from f(−x) = −x
3
− 3x
2
+ 6x + 8).
Polynomial functions are usually evaluated most compactly using nested multiplication.
(This is sometimes referred to as Horner's method.) As an illustration, the function from the
previous example can be rewritten as
f(x) = [(x − 3)x − 6]x + 8 .
This representation is more easily programmed and more efficiently executed than the
original form, especially since _ fills the stack with the value of x.
Example: During the winter of '78, Arctic explorer Jean-Claude Coulerre, isolated at his
frozen camp in the far north, began scanning the southern horizon in anticipation of the sun's
reappearance. Coulerre knew that the sun would not be visible to him until early March,
when it reached a declination of 5° 18'S. On what day and time in March was the chilly
explorer's vigil rewarded?
The time in March when the sun reached 5° 18'S declination can be computed by solving the
following equation for t:
D = a
4
t
4
+ a
3
t
3
+ a
2
t
2
+ a
1
t + a
0
Where D is the declination in degrees, t is the time in days from the beginning of the month,
and
a
4
= 4.2725 × 10
−8
a
3
= −1.9931 × 10
−5
a
2
= 1.0229 × 10
−3
a
1
= 3.7680 × 10
−1
a
0
= −8.1806 .
This equation is valid for 1 ≤ t < 32, representing March, 1978.