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:

• 3x + 5y = 7

• 6(x - 5) ² + 24x

• 6x ³ + 5x ² - 8x + 7

• (3x ² + 7x - 1) ⁴

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:

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 ²y’ 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 ²y 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 ²y, 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.