Civil Engineering Reference
In-Depth Information
Refer to Table 1.17 for the basic procedure for the root squaring. Table 1.18
shows the process for this polynomial.
To determine the proper sign of the roots from the Enke roots, a check
is required.
1
128
1
≅=
()
77
rb
1 158 10
.
=±
4 000
.
m
1
1
1
128
1
(
)
()
138
1 365 10
.
=
b
b
m
2
r
=
=±
3 000
.
2
77
1 158 10
.
1
1
128
1
(()
()
176
=
m
4 646 10
.
b
b
3
r
=
=±
2 000
.
3
138
1 365 10
.
2
1
128
1
()
()
176
=
m
4 646
10
4 646 10
.
b
b
r
4
=
=±
1 000
.
4
176
.
3
Substituting the Enke roots into the original equations yields
x
1
=
4
.
000,
x
2
=
3
.
000,
x
3
=
2
.
000,
and
x
4
=
1
.
000
.
1.12.2
REAL AND EQUAL ROOTS
After many cycles of squaring the polynomial, the second possible solu-
tion type will occur when the coefficients of the derived polynomial are
the squares of the terms in the preceding cycle with the exception of one
term that is ½ the square of the term in the preceding cycle. This indicates
that two of the roots are equal, the one with the ½ squared term and the
next one to the right. Furthermore, if one term is
1
/
3
the square of the term
in the proceeding cycle, three of the roots are equal—the term with the
1
/
3
squared term and the next two to the right. A similar relationship will
occur if four or more roots are equal. The roots (Enke roots) will have a
relationship similar to the following assuming
r
1
= r
2
and considering only
the dominant terms in each expression:
mmmmm m
b
=++=+∴≅
=
r
r
r
r
r br
2
1
1
2
3
1
1
1
1
brr
mm mm mm m
+
r r
+
r r
=
r r
m
∴≅
br
2
m
2
1
2
1
3
2
3
1
1
2
1
mmm mmm
2
mm
b rrr rrr brr
=
= ∴≅
3
1 23 113
3
1
3
