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, 2a, 3b, 7ab 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: +2a, -58, -7bc, +3xy, -28a2bc
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 +2a, -7bc, 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 -72a3, 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 +1ab. When a term has a minus sign but no coefficient, it is assumed that the coefficient is -1 (minus one). For example, –cd means -1cd.
Terms can be combined in expressions such as the following: 2a – 3b + 8c. The parts of an expression joined by the signs + and – are the terms. Thus this expression consists of three terms, that is, +2a, -3b, +8c.
Another name for a term ismonomial. An expression consisting of two monomials, or two terms, is called a binomial. An example is 5ab – 7cd. An expression consisting of three terms is called a trinomial, such as 7a – 8b + 3c. Any expression of more than one term is called apolynomial.
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 2a and 5a, 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 3ab, -4ab, 100ab, -1.5ab 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 5x2, 7x2, -12x2, 150x2 are also like terms, because they all contain the letter x to the second power.
Unlike terms contain different letters, or have different exponents of the letters. The following are groups of unlike terms.
2a, 2b, 2c
2ab, 5ac, 3bc
2a2b, 2a3b, 3ab, 7ab2, 4ab3, 5a2b2
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, x2yz3 = x2z3y = yx2z3 = yz3x2 = z3x2y = z3yx2
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.