Civil Engineering Reference
In-Depth Information
1.4
SYntHEtic DiViSiOn
Synthetic division is taught in most algebra courses. The main outcome is
to divide a polynomial by a value,
r
. This is in fact the division of a poly-
nomial,
f
(
x
) = 0 by the linear equation
x - r
. The general polynomial can
be divided by
x − r
as follows:
()
=
n
n
−
1
n
−
2
n
−
3
1
fx ax ax ax
+
+
+ ++ +=
ax
ax a
0
0
1
2
3
n
−
1
n
Table 1.1.
Synthetic division
r
a
0
a
1
a
2
…….
a
n-1
a
n
0
rb
1
rb
2
…….
rb
n-1
rb
n
b
1
b
2
b
3
…….
b
n
R
The results,
b
, are the sum of the rows above (i.e.,
b
1
=
a
0
+
0
or
b
n
=
a
n-
1
+
rb
n-
1
). If
r
is a root, then the remainder,
R
, will be zero. If
r
is not
a root, then the remainder,
R
, is the value of the polynomial for
f
(
x
) at
x
=
r
.
Furthermore, after the first division of a polynomial, divide again to
find the value of the first derivative equal to the remainder times one fac-
torial,
R*1!.
After the second division of a polynomial, divide again to
find the value of the second derivative equal to the remainder times two
factorial,
R*2!.
Continuing this process, and after the third division of a
polynomial, divide again to find the value of the third derivative equal to
the remainder times three factorial,
R*3!
. Basically, two synthetic divi-
sions yield the first derivative, three synthetic divisions yield the second
derivative, four synthetic divisions yield the third derivative, and so on.
Example 1.5 Synthetic division
Find
f
(1) or divide the following polynomial by
x
-
1.
3
2
6160
x
−+−=
x
x
Set up the equation as shown below by writing the divisor,
r
, and coeffi-
cient,
a
, in the first row.
Table 1.2.
Example 1.5 Synthetic division
1
1
11
−
6
−
6
