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
, 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
, or a quantity
whose value is fixed and may not be changed during a particular discussion.
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
, 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
. 2x + 3y, (x
y + 2xy
) – x/y +
x , and
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
of the algebraic
expression. An algebraic expression consisting of just one term is called a
if it is composed of two terms; a
if it has three terms; and in general, a
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
of the others. For example, in
is the coefficient
is the coefficient of
is the coefficient of
. At times, we need to distinguish
(letter-symbol) coefficients. In the given example,
numerical coefficient while
is the literal coefficient. Terms that have the same literal
coefficients are called
Time to think
Draw a Venn diagram describing the relationship among the sets of numbers
Define the operations that can be defined on each set above and the underlying