1.4. Ring of Polynomials
We have seen from the past sections that the basic objects of Algebra are numbers. We
have several types of numbers ranging from counting or natural, to integers, rational, irrational,
real and complex numbers. A set of numbers may have an underlying algebraic structure
depending on the operations defined on them.
The world of mathematics will be small if only numbers are to be considered as its
objects. At times, quantities have no definite values while there are occasions when a particular
value is often used so that a symbol for it is found very useful. The first type of quantities is
called
variables
, or those which are represented only by letters and whose values may be
arbitrarily chosen depending on the situation. The second type is called a
constant
, or a quantity
whose value is fixed and may not be changed during a particular discussion.
Illustration 1.
In the formula, F = ma, relating force to the product of mass and acceleration, the
variables are m (mass), a (acceleration) and F (force). In the expression, ½ gt
2
, t is the variable
while g is a constant referring to the value of gravity. Of course, ½ is also a constant.
The use of symbols in representing quantities (or numbers) led to the notion that algebra
is generalized arithmetic.
Any combination of numbers and symbols related by the operations we have described in
the earlier sections will be called an
algebraic expression
.
Illustration 2
. 2x + 3y, (x
2
y + 2xy
2
) – x/y +
x , and
π
x
3
+ 2
x
are algebraic expressions.
Any algebraic expression consisting of distinct parts separated by plus or minus signs is
called an algebraic sum. Each distinct part, together with its sign, is called a
term
of the algebraic
expression. An algebraic expression consisting of just one term is called a
monomial
; a
binomial
,
if it is composed of two terms; a
trinomial
if it has three terms; and in general, a
multinomial
if it
has any number of terms.
A particular term of an algebraic expression is composed of one or more factors. Each of
the factors may be called the
coefficient
of the others. For example, in
3u
4
v
,
3
is the coefficient
of
u
4
v
,
3u
4
is the coefficient of
v
and
3v
is the coefficient of
u
4
. At times, we need to distinguish
between
numerical
and
literal
(letter-symbol) coefficients. In the given example,
3
is the
numerical coefficient while
u
4
v
is the literal coefficient. Terms that have the same literal
coefficients are called
similar terms
.
Time to think
!
1)
Draw a Venn diagram describing the relationship among the sets of numbers
enumerated above.
2)
Define the operations that can be defined on each set above and the underlying
algebraic structures.