**Definition of algebraic terms**

You have already learned earlier that an algebraic quantity is indicated by letter. An algebraic quantity can, however, be indicated by a number or a combination of a number and a letter or letters. For example, 34, 2*a*, 3*b*, 7*ab* are quantities. If a quantity consists of a sign, a number, and a letter or several letters, it is called a *term. *For example, the following quantities are terms: +2*a*, -58, -7*bc*, +3*xy*, -28*a ^{2}bc*

A term with a plus sign is a *positive term* and one with a minus sign is a *negative term. *If a term consists of a numeral only, it is called *numerical term*, or a *constant*; if a term consists only of letters, or of letters and numerals, it is called a *literal term. *For example, +12.8 and -58 are numerical terms, and +2*a*, -7*bc*, and *ab/c* are literal terms. The numeral in front of a term is called *coefficient*, the letter or letters of the term are called *literal parts*, and the small number above and to the right of a letter is called an *exponent. *For example, in the term -72*a ^{3}, *the number -72 is the

*coefficient*,

*a*is the

*literal part*, and 3 is the

*exponent*of the letter

*a.*

When a term has neither a sign nor a coefficient, it is understood that the coefficient is +1 (plus one). For example, *ab* means +1*ab*. When a term has a minus sign but no coefficient, it is assumed that the coefficient is -1 (minus one). For example, –*cd* means -1*cd. *

Terms can be combined in expressions such as the following: 2*a* – 3*b* + 8*c. *The parts of an expression joined by the signs + and – are the terms. Thus this expression consists of three terms, that is, +2*a*, -3*b*, +8*c.*

Another name for a term is*monomial. *An expression consisting of two *monomials*, or two terms, is called a *binomial*. An example is 5*ab* – 7*cd. *An expression consisting of three terms is called a trinomial, such as 7*a* – 8*b* + 3*c. *Any expression of more than one term is called a*polynomial.*

**Like and Unlike Terms**

Terms that contain the same letters with the same exponents are called *like* terms. It is very important to recognize like terms in algebraic expressions because *only like terms can be added or subtracted. *Terms 2*a* and 5*a*, for example, are like terms because they contain the same literal part, or letter *a. *Note that the coefficient 2 and 5 are different. Like terms do not have the same value but must contain the same letter or letters with the same exponents.

Terms 3*ab*, -4*ab*, 100*ab*, -1.5*ab* are also like terms because they all contain the same letters. Each one has, of course, different value, because the coefficients and signs are different.

Terms 5*x ^{2}*, 7

*x*, -12

^{2}*x*, 150

^{2}*x*are also like terms, because they all contain the letter

^{2}*x*to the second power.

*Unlike* terms contain different letters, or have different exponents of the letters. The following are groups of *unlike* terms.

2*a*, 2*b*, 2*c*

2*ab*, 5*ac*, 3*bc*

2*a ^{2}b*, 2

*a*, 3

^{3}b*ab*, 7

*ab*, 4

^{2}*ab*, 5

^{3}*a*

^{2}b^{2}
In the last group you see the *same letters*, but their *exponents* are different; therefore they are *unlike* terms.

Always consider the letters and their exponents when deciding which are the like terms. One fact to remember is this: *The sequence or order of letters in a term is not important. *The term *xy* is the same as the term *yx. *Similarly, the following terms are *like* terms:

*abc = acb = bac = bca = cab = cba *

Also, *x ^{2}yz^{3} = x^{2}z^{3}y = yx^{2}z^{3} = yz^{3}x^{2 }= z^{3}x^{2}y = z^{3}yx^{2}*

Keep this fact in mind when working with like terms.

It is customary to write the letters in a term in *alphabetical order*, but that is not absolutely necessary. The letters can be written in any sequence; this will not change the term in any way. But when looking for like terms, keep this point in mind: *Like terms contain the same letters, each with the same exponent.*