expression 5abc there are four factors, the numeral factor 5, and the three literal factors, a, b, and c. 9. The sign of division (÷) is read "divided by." When written between two quantities, it indicates that the first is to be divided by the second. There are three signs used to denote division. α a+b,, alb, denote that a is to be divided by b. Thus, When 10. The sign of equality (=) is read "equal to." written between two quantities, it indicates that they are equal to each other. Thus, the expression a+b=c indicates that the sum of a and b is equal to c. If a stands for 3, and b for 5, c will be equal to 8. 11. The sign of inequality (> or <) is read "greater than " or "less than." When placed between two quantities, it indicates that they are unequal, the greater one being placed at the opening of the sign. Thus, the expression ab indicates that a is greater than b, and the expression c<d indicates that c is less than d. 12. The sign.. means "therefore," or "consequently." 13. A coefficient is a number written before a quantity, to show how many times the quantity is taken additively. Thus, in the expression a+a+a+a+a=5a, 5 is the coefficient of a. A coefficient may be denoted either by a number or by a letter. Thus, 5x indicates that x is taken 5 times, and ax indicates that x is taken a times. If no coefficient is written, the coefficient 1 is understood. Thus, a is the same as la. 14. An exponent is a number written at the right and a little above a quantity, to indicate how many times the quantity is taken as a factor. Thus, and the 2, 3, and 4 are exponents. The expressions are read, a square,' a cube" or "a third," a fourth;" and if we have am, in which a enters m times as a factor, it is read " α to the mth," or simply "a mth." The exponent 1 is generally omitted. Thus, a is the same as a, each denoting that a enters but once as a factor. 15. A power is a product which arises from the multiplication of equal factors. Thus, 16. A root of a quantity is one of the equal factors. The radical sign (√), when placed over a quantity, indicates that a root of that quantity is to be extracted. The root is indicated by a number written over the radical sign, called an index. When the index is 2, it is generally omitted. Thus, Va, or Va, indicates the square root of a. Va indicates the fourth root of a. Wa indicates the mth root of a. 17. An algebraic expression is a quantity written in algebraic language. Thus, 3a is the algebraic expression of three times the number denoted by a; 5a2, that of five times the square of a; 7a3b2, that of seven times the cube of a multiplied by the square of b; 3a-5b, that of the differ ence between three times a and five times b; and 2a2-3 ab +462, that of twice the square of a, diminished by three times the product of a by b, augmented by four times the square of b. 18. A term is an algebraic expression that can be written without the aid of either of the signs or -. Thus, 3a, 2ab, 5a2b2, are terms. A term may be preceded by either of the signs + or In this case the sign is called the sign of the term, and is used to show the sense in which the term is taken. Thus +3 a shows that 3 a is taken positively, and — 5 a2b2 shows that 5a2b is taken negatively. 19. The degree of a term is the number of its literal factors. Thus, 3a is a term of the first degree, because it contains but one literal factor; 5a2 is a term of the second degree, because it contains two literal factors (the factors in this case are equal); 7ab is a term of the fourth degree, because it contains four literal factors. The degree of a term is determined by the sum of the exponents of all its letters. 20. A monomial is a single term, unconnected with any other by the signs + or Thus, 3a2, 363 a, are monomials. 21. A polynomial is a collection of terms connected by the signs or; as, 3a -5, or 2a3 — 3b+4 b2. 22. A binomial is a polynomial of two terms; as, a+b, 3a2- c2, 6ab - c2. 23. A trinomial is a polynomial of three terms; as, abc-a3 c3, ab - gh-f. 24. Homogeneous terms are those which are of the same degree. Thus, the terms abc, -a, c3, are homogeneous; as are the terms ab, - gh. 25. A polynomial is homogeneous when all its terms are homogeneous. Thus, the polynomial abc-a+c is homogeneous, but the polynomial ab-gh-f is not homogeneous. 26. Similar terms are those which have a common unit; that is, have a common literal part. Thus, 7 ab+3ab -2ab are similar terms, and so also are 4a2b2 - 2a2b2 — 3a2b2; but the terms of the first polynomial and of the last are not similar. 27. The vinculum, and the brackets, [], terms which are to be of the expressions απ ; the bar, ; the parentheses, (); are each used to connect two or more operated upon as a whole. Thus, each a+b+cxx, +b (a+b+c) × x, + c [a+b+c] × x, indicates that the sum of a, b, and c, is to be multiplied by x. 28. The reciprocal of a quantity is 1 divided by that quantity. Thus, 29. The numerical value of an algebraic expression is the result obtained by assigning a numerical value to each letter, and then performing the operations indicated. Thus, the numerical value of the expression ab+be+d, when a = 1, b=2, c=3, and d=4, is 1x2+2x3+4=12, by performing the indicated operations. EXERCISES IN WRITING ALGEBRAIC EXPRESSIONS. 1. Write a added to b. Ans. a+b. 2. Write b subtracted from a. Ans. a-b. 3. Six times the square of a, minus twice the square of b. 4. Six times a multiplied by b, diminished by 5 times c cube multiplied by d. 5. Nine times a, multiplied by e plus d, diminished by 8 times b multiplied by d cube. 6. Five times a minus b, plus 6 times a cube into b cube. 7. Eight times a cube into d fourth, into c fourth, plus 9 times c cube into d fifth, minus 6 times a into b, into c square. 8. Fourteen times a plus b, multiplied by a minus b, plus 5 times a, into c plus d. 9. Six times a, into c plus d, minus 5 times b, into a plus c, minus 4 times a cube b square. 10. Write a, multiplied by c plus d, plus ƒ minus g. 11. Write a divided by b+c in three ways. 12. Write a b divided by a+b. 13. Write a polynomial of three terms, of four terms, of five, of six. 14. Write a homogeneous binomial of the first degree, of the second, of the third, fourth, fifth, sixth. 15. Write a homogeneous trinomial of the first degree, with its second and third terms negative; of the second degree; of the third; of the fourth. 16. Write in the same column, on the slate or blackboard, a monomial; a binomial; a trinomial; a polynomial of four terms, of five terms, of six terms, and of seven terms; and all of the same degree. |