Civil Engineering Reference
In-Depth Information
algebra. Examples of such are exponential functions, trigonometric func-
tions, and the inverses of each. The following are examples of transcen-
dental equations:
()
+
()
=
+=
cosx sinx
e
x
0
15
0
Transcendental functions may have an infinite number of roots or may not
have any roots at all. For example, the function
sin
(
x
)
=
0 has an infinite
number of roots
x =
±
kx
and
k
= 0, 1, 2
….
The solution of algebraic or transcendental equations is rarely carried
out from the beginning to end by one method. The roots of the equation
can generally be determined by one method with some small accuracy,
and then made more accurate by other methods. For the intent and pur-
pose of this text, only a handful of the available methods are discussed.
These methods include: Descartes' Rule, Synthetic Division, Incremental
Search, Refined Incremental Search, Bisection, False Position, Secant,
Newton-Raphson, Newton's Second Order, Graeffe's Root Squaring, and
Bairstow's methods. Some of these methods are used to solve specific
types of equations, while others can be used for both equation types.
1.2 POLYnOMiALS
A
polynomial
is defined as an algebraic equation involving only positive
integer
(whole number) powers of
x
. Polynomials are generally expressed
in the following form:
n
n
−
1
n
−
2
n
−
3
1
ax ax ax
+
+
+
ax
+
+
ax a
+
=
0
0
1
2
3
n
−
1
n
In most cases, the polynomial form is revised by dividing the entire equa-
tion by the coefficient of the highest power of
a
,
a
0
, resulting in the fol-
lowing form:
n
n
−
1
n
−
2
n
−
3
1
xax ax
+
+
+
a x
+
+
a xa
+
=
0
1
2
3
n
−
1
n
For these polynomials, the following apply:
• The order or degree of the polynomial is equal to the highest power
of
x
and the number of roots is directly equal to the degree or
n
,
where
a
n
is not equal to 0
.
For example, a sixth degree polynomial,
or a polynomial with
n
= 6 has six roots.
• The value of
n
must be a non-negative integer. In other words, it
must be a whole number that is equal to zero or a positive integer.
