Terminology of Algebraic Expressions
In this note we give brief descriptions and illustrations of
some names of things encountered in algebra.
An algebraic expression is a sequence of
numbers and literal symbols together with arithmetic operation
symbols and perhaps pairs of brackets. When the literal symbols
are replaced by actual numbers, it should be possible to reduce
the resulting numerical expression to a single numerical value.
Examples of algebraic expressions are:
 6x^{ 3} + 5x^{ 2}  8x + 7
Algebraic expressions consist of one or more terms. The terms
of the expression are the parts of the expression which are
separated by ‘+’ or ‘‘ signs.
Thus, the expression:

has 3 terms 

has 2 terms 

has 1 term 
 6x^{ 3} + 5x^{ 2}  8x + 7

has 4 terms 

has 1 term 
Terms can be products of two or more factors.
A product results when two or more quantities
are multiplied together. The quantities being multiplied together
to form a product are called its factors. Thus
 ‘3x’ is the product of two factors,
‘3’ and ‘x’; or ‘3’ and
‘x’ are the factors of the product
‘3x’
 ‘6x^{ 2}y’ is the product of three
factors: ‘6’, ‘x^{ 2}’, and
‘y’
Of course, ‘x^{ 2}’ could itself be regarded
as the product of two factors: ‘x’ and ’x’.
In that case, 6x^{ 2}y becomes the product of four
factors.
Often a term is a product of a constant or number and a part
which is a literal symbol or a product or two or more literal
symbols. The numerical factor is often referred to as the coefficient
or numerical coefficient of the term. Thus
 the term ‘3x’ has the numerical coefficient
‘3’
 the term ‘6x^{ 2} y’ has the numerical
coefficient ‘6’,
etc.
The word coefficient is also used more
generally to refer to a factor or group of factors in a term.
Thus, for example, in the term 6x^{ 2} y,
‘6x^{ 2} ’ is the coefficient of
‘y’
and
‘y’ is the coefficient of ‘6x^{ 2}
’
Terms which have identical symbolic parts are said to be
like terms. Logically, two terms with different symbolic
parts would be called unlike terms. So
 3x^{ 2} y and 7x^{ 2} y are like terms
(the symbolic part in both cases is ‘x^{ 2}
y’)
but
 3x^{ 2} y and 7xy^{ 2} are unlike terms,
because the symbolic part of the first one is x^{ 2}
y, which is different from xy^{ 2} , the symbolic
part of the second one.
There are specific names for expressions which indicate how
many terms they contain:
monomials are expressions with just one term
for example: 5, 6x^{ 2}y, 3x, 7xyz, etc. are monomials
binomials are expressions with two terms:
for example: x + y, 3x^{ 2} – 4y, 5 + 2x, etc.,
are binomials
trinomials are expressions with 3 terms
for example
5x^{ 2} + 2xy 3y^{ 2 }is a trinomial
is a trinomial
multinomials are expressions with several
terms
polynomials are expressions with two or more
terms in which the symbolic part of each term is a power of a
single symbol (the same one in all terms). The degree of
a polynomial is the highest power that it contains.
for example:
3x^{ 2} + 2x + 5 is a polynomial of degree 2 (or, a
polynomial of the second degree)
5x^{ 7} + 3x^{ 4} + 9x^{ 3 } 7x^{
2} + 2 is a polynomial of degree 7.
