HP HP-15C

Section 1: Using ["SOLVE | Effectively 11
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
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 /( ~x) = — x3 — 3x'2 +6x + 8).
Polynomial functions are usually evaluated most compactly using
nested multiplication. (This is sometimes referred to as Homer'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 | SOLVE)
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 = a4t4 + a3t3 + a2t2 + alt + a0
where D is the declination in degrees, t is the time in days from the
beginning of the month, and