234 Appendix D: A Detailed Look at _
lines should subtract the known root (to 10 significant digits) from the x-
value and divide this difference into the function value. In many cases the
root will be a simple one, and the new function will direct _ away
from the known root.
On the other hand, the root may be a multiple root. A multiple root is one
that appears to be present repeatedly, in the following sense: at such a
root, not only does the graph of f (x) cross the x-axis, but its slope (and
perhaps the next few higher-order derivatives) also equals zero. If the
known root of your equation is a multiple root, the root is not eliminated
by merely dividing by the factor described above. For example, the
equation
f (x) = x(x − a)
3
= 0
has a multiple root at x = a (with a multiplicity of 3). This root is not
eliminated by dividing f (x) by (x − a). But it can be eliminated by dividing
by (x − a)
3
.
Example: Use deflation to help find the roots of
60x
4
− 944x
3
+ 3003x
2
+ 6171x − 2890 = 0.
Using Horner’s method, this equation can be rewritten in the form
{[(60x − 944)x + 3003]x + 6171}x − 2890 = 0.
Program a subroutine that evaluates the polynomial.
Keystrokes
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¥
000-
´
CLEAR
M
000-
´
b
2
001-42,21, 2
6
002- 6
0
003- 0
*
004- 20
9
005- 9
4
006- 4
4
007- 4
-
008- 30
*
009- 20
3
010- 3
011- 0
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