signs + or are called its terms. When an expression consists of two terms it is called a binomial expression; when it consists of three terms it is called a trinomial expression; any expression consisting of several terms may be called a multinomial expression or a polynomial expression. When an expression does not contain parts connected by the sign + or the sign - it may be called a simple expression, or it may be said to contain only one term.

Thus abc is a simple expression; abc + x is a binomial expression, of which abc is one term, and x is the other; ab + ac-bc is a trinomial expression, of which ab, ac, and be are the terms.

14. When one number consists of the product of two or more numbers, each of the latter is called a factor of the product. Thus b and c are factors of the product abc.

a,

15. A product may consist of one factor which is a number represented arithmetically, and of another factor which is a number represented algebraically, that is, by a letter or letters; in this case the former factor is said to be the coefficient of the latter. Thus in the product 7abc the factor 7 is called the coefficient of the factor abc. Where there is no arithmetical factor, we may supply unity; thus we may say that, in the product abc, the coefficient is unity.

And when a product is represented entirely algebraically, any one factor may be called the coefficient of the product of the remaining factors. Thus, in the product abc, we may call a the coefficient of bc, or b the coefficient of ac, or c the coefficient of ab. If it be necessary to distinguish this use of the word coefficient from the former, we may call the latter coefficients literal coefficients, and the former coefficients numerical coefficients.

16. If a number be multiplied by itself any number of times, the product is called a power of that number. Thus a xa is called the second power of a; also a xaxa is called the third power of a; and a×a×a×a is called the fourth power of a; and so on.

above the quantity the number which represents how often it is repeated in the product. Thus a2 is used to express a × a; also

of a

as is used to express a xa xa; and at is used to express a xa xa xa; and so on. And a' may be used to denote the first power or a itself; that is, a' has the same meaning as a.

Numbers placed above a quantity to express the powers of that quantity are called indices of the powers, or exponents of the powers; or more briefly indices or exponents.

18. Thus we may sum up the two preceding articles as follows: a×a×a× &c. to n factors is expressed by a”, and n is called the index or exponent of a”, where n may denote any whole number.

19. The second power of a or a2 is often called the square of α, and the third power of a or a3 is often called the cube of a. The symbol a is read thus "a to the fourth power" or riefly "a to the fourth;" and a" is read thus "a to the nth "

20. The square root of any assigned number is that number hich has the assigned number for its square or second power. "he cube root of any assigned number is that number which has ne assigned number for its cube or third power. The fourth root any assigned number is that number which has the assigned umber for its fourth power. And so on.

21. The square root of a number a is denoted thus a, or mply thus a. The cube root of a is denoted thus /a. The irth root of a is denoted thus a. And so on.

The sign is said to be a corruption of the initial letter of e word radix. This sign is sometimes called the radical sign.

22. Terms are said to be like or similar when they do not fer at all or differ only in their numerical coefficients; otherwise y are said to be unlike. Thus 4a, 6ab, 9a2 and 3a2bc are pectively similar to 15a, 3ab, 12a2 and 15a2bc. And ab, a'b,

is called a dimension of the product, and the number of the letters is the degree of the product. Thus a b3c or a xa xbxbxbx c is said to be of six dimensions or of the sixth degree. A numerical coefficient is not counted; thus 9a3b and a3b are of the same dimensions, namely of seven dimensions. Thus the degree of a term or the number of dimensions of a term is the sum of the exponents, provided we remember that if no exponent is expressed the exponent 1 must be understood as indicated in Art. 17.

24. An algebraical expression is said to be homogeneous when all its terms are of the same dimensions. Thus 7a3+ 3a2b+ 4abc is homogeneous, for each term is of three dimensions.

The following examples will serve for an exercise in the preceding definitions.

EXAMPLES.

If a=1, b=3, c=4, d=6, e2 and f=0, find the numerical values of the following twelve algebraical expressions:

13. Find the value of (9-y) (x + 1) + (x + 5) (y + 7) − 112, when x= 3 and y=5.

14. Find the value of x /(x2 – 8v) + v /(x2 + 8v), when x-5

17. Any power of a quantity is usually expressed by placing above the quantity the number which represents how often it is repeated in the product. Thus a2 is used to express a × a; also a3 is used to express a × a × a; and a* is used to express a ×a×a×a; and so on. And a' may be used to denote the first power of a or a itself; that is, a' has the same meaning as a.

Numbers placed above a quantity to express the powers of that quantity are called indices of the powers, or exponents of the powers; or more briefly indices or exponents.

18. Thus we may sum up the two preceding articles as follows: a×a×a× &c. to n factors is expressed by a”, and n is called the index or exponent of a", where n may denote any whole number.

19. The second power of a or a2 is often called the square of a, and the third power of a or a3 is often called the cube of a. The symbol a is read thus "a to the fourth power” or briefly "a to the fourth;" and a" is read thus "a to the nth"

20. The square root of any assigned number is that number which has the assigned number for its square or second power. The cube root of any assigned number is that number which has the assigned number for its cube or third power. The fourth root of any assigned number is that number which has the assigned number for its fourth power. And so on.

a, or

21. The square root of a number a is denoted thus simply thus a. The cube root of a is denoted thus /a. The fourth root of a is denoted thus a. And so on.

The sign is said to be a corruption of the initial letter of the word radix. This sign is sometimes called the radical sign.

22. Terms are said to be like or similar when they do not differ at all or differ only in their numerical coefficients; otherwise they are said to be unlike. Thus 4a, 6ab, 9a2 and 3a2bc are respectively similar to 15a, 3ab, 12a2 and 15a2bc. And ab, a3b, ab2 and abc are all unlike.

if a represent 10, b represent 6, and c represent 5, then

a+b-c=a- c + b = b − c + α.

If however a represent 2, b represent 6, and c represent 5, then the expression a -c+b presents a difficulty because we are thus apparently required to take a greater number from a less, namely 5 from 2. It will be convenient to agree that such an expression as a - c+b when c is greater than a shall be understood to mean the same thing as a + b − c. At present we shall never use such an expression except when c is less than a + b, so that a + b − c presents no difficulty. Similarly we shall consider − b + a to mean the same thing as a-b. We shall recur to this point hereafter in Chapter V.

28.

Thus the numerical value of an expression remains the same whatever may be the order of the terms which compose it. This, as we have seen, follows, partly from our notions of addition and subtraction, and partly from an agreement as to the meaning we ascribe to an expression when our ordinary arithmetical notions are not strictly applicable. Such an agreement is called in Algebra a convention, and conventional is the corresponding adjective.

29. We shall frequently, as in Article 26, have to distinguish the terms of an expression which are preceded by the sign + from the terms which are preceded by the sign -, and thus the following definition is adopted. The terms in an algebraical expression which are preceded by no sign or which are preceded by the sign + are called positive terms; the terms which are preceded by the sign are called negative terms. This definition is introduced merely for the sake of brevity, and no meaning is to be given to the words positive and negative beyond what is expressed in the definition. The student will notice that terms preceded by no sign are treated as if they were preceded by the sign +.

30. Sometimes an expression includes several like terms; in this case the expression admits of simplification For example

20