Civil Engineering Reference
In-Depth Information
Since the remainder,
R
, is zero,
f
(3) = 0
and
r
= 3 is a root. The polynomial
is now written as:
(
)
=−
(
)
−+
(
)
(
)
(
)
=
3
2
2
x
−+−= −
616 1
x
x
x
x
5
x
6
x
1
x
−
2
x
−
3
0
The roots are
x
= 1, 2, 3.
Example 1.6 Synthetic division
Find
f
(
-
1),
f
′(
-
1), and
f
″(
-
1) or perform three divisions of the following
polynomial by
x +
1:
x
3
−+−=
2
6160
x
x
Set up the equation as shown in the following by writing the divisor,
r
, and
coefficient,
a
, in the first row.
Table 1.6.
Example 1.6 Synthetic division
−
1
1
−
6
11
−
6
Add the columns by starting at the left. Multiply each result by
r
= 1 and
add this to the next column.
Table 1.7.
Example 1.6 Synthetic division
−
1
1
−
6
11
−
6
0
−
1
7
−
18
1
−
7
18
−
24
Since the remainder,
R
, is
-
24,
f
(
-
1) =
-
24,
the polynomial evaluated at
-
1
is
-
24. Performing a check as follows:
f
()
=
()
−
()
+
()
−=−
3
2
1
1 61 11 16 24
Now divide the remaining polynomial again by
-
1 to find
f
′(
-
1).
