Ruffini-Horner Method for a Polynomial in Powers of x-h

​
degree
n
2
3
4
5
shift
h
-3
-2
-1
1
2
3
4
new polynomial
steps
k
0
1
2
3
4
P(x)4
4
x
-
3
x
+3x-2
4
-1
0
3
-2
4
3
3
6
4
3
3
6
4
​
This Demonstration shows the transformation of a polynomial in powers of
x
into a polynomial in powers of
x-h,
using the Ruffini–Horner method.

Details

Given a polynomial
P(x)=
a
n
n
x
+
a
n-1
n-1
x
+…+
a
1
x+
a
0
,
find a way to express it as a polynomial in
x-h
:
n
b
n
(x-h)
+
n-1
b
n-1
(x-h)
+…+
b
1
(x-h)+
b
0
.
One method is to use a Taylor series
P(x)=P(h)+
′
P
(h)(x-h)+⋯+
(n-1)
P
(h)
(n-1)!
n-1
(x-h)
+
(n)
P
(h)
n!
n
x
.
Another way is to make use of synthetic division, discovered by Ruffini in 1804 and Horner in 1819.

References

[1] Wikipedia. "Paolo Ruffini." (Dec 12, 2016) en.wikipedia.org/wiki/Paolo_Ruffini.
[2] Wikipedia. "William George Horner." (Dec 12, 2016) en.wikipedia.org/wiki/William_George_Horner.

External Links

Horner's Method
Synthetic Division (Ruffini's Rule)
Horner's Rule (Wolfram MathWorld)
Ruffini's Rule (Wolfram MathWorld)

Permanent Citation

Izidor Hafner
​
​"Ruffini-Horner Method for a Polynomial in Powers of x-h"​
​http://demonstrations.wolfram.com/RuffiniHornerMethodForAPolynomialInPowersOfXH/​
​Wolfram Demonstrations Project​
​Published: December 14, 2016