Civil Engineering Reference
In-Depth Information
1.12.1
REAL AND DISTINCT ROOTS
The first possible solution type will occur after many cycles of squaring
the polynomial; the coefficients of the derived polynomial are the squares
of the terms from the preceding cycle. This is known as the regular solu-
tion and yields real and
distinct roots
(not equal). The roots of the polyno-
mial or the derived polynomials can be determined from the factored form
of the polynomials. The Enke roots of a polynomial are the negatives of
the roots of the polynomial. If
r
is denoted as the Enke root designation,
then
x
1
=
−
r
1
,
x
2
=
−
r
2
… x
n
=
−
r
n
.
The third degree polynomial is shown
in factored form:
()
==+++
()
== −
3
2
fx xaxaxa
fx xxxxxx
0
0
1
2
3
(
)
(
)
(
)
−
−
1
2
3
If the previous equation is multiplied out, the following is the result:
()
==−++
(
)
(
)
−
fx xxxxx xxxxxxxxx
0
3
2
+
+
+
1
2
3
12 13 23
123
Therefore, for the polynomial, the original coefficients are as follows:
(
)
a x xx
axxxxxx
a
=− ++
1
1
2
3
=++
=−
2
1 2
1 3
2 3
x xx
3
1 23
The Enke roots of
x
1
=
−
r
1
,
x
2
=
−
r
2
, and
x
3
=
−
r
3
are substituted in. The
sign has been lost so the Enke roots are used as the basis (
x
1
=
−
r
1
,
x
2
=
−
r
2
,
etc.), then the following is true:
arrr
a r rrr
a rr
=++
=++
=
1
1
2
3
2
1 2
1 3
2 3
3
1 23
As the cycles (m) continue, the derived polynomial becomes the following:
()
==+++
fx ybybyb
0
3
2
1
2
3
