XV. The following characters are used to connect several quantities together, viz.:— Thus m+n.x, or (m+n) x signifies that the quantity denoted by m+n is to be multiplied by x, and {+}{} signifies that + is to be multiplied by ።+ XVI. The signs .. therefore or consequently, and . because, are used to avoid the too frequent repetition of these words. XVII. Every number written in algebraic language, that is, by aid of algebraic symbols, is called an algebraic quantity, or, a literal quantity, or, an algebraic expression. Thus 3 a is the algebraic expression for three times the number a; 5 a2 is the algebraic expression for five times the square of the number a; 7 a3 b3 is the algebraic expression for seven times the fifth power of a multiplied by the cube of b. 3 a2-6 b3 c1 is the algebraic expression for the difference between three times the square of a and six times the cube of b multiplied by the fourth power of c. 2a-3b2 c3+4 d e f is the algebraic expression for twice a, diminished by three times the square of b multiplied by the cube of c and augmented by four times the fourth power of d multiplied by the product of the fifth power of e and the sixth power of f. XVIII. An algebraic quantity, which is not combined with any other by the sign of addition or subtraction, is called a monomial, or, a quantity of one term, or simply, a term. Thus, 3 a2, 4 b2, 6 c, are monomials. An algebraic expression, which is composed of several terms, separated from each other by the signs + or - - is called generally a polynomial. Thus, 3a2+4b2-6c+d, is a polynomial. A polynomial, consisting of two terms only, is usually called a binomial, when consisting of three terms, a trinomial. Thus, a + b, 3 b2 c — x z, are binomials, and a + b· · c, 3 m2 n3 — 6 p3r +9 d, are trinomials. XIX. The numerical value of an algebraic expression is the number which results from giving particular values to the letters which compose the expression, and performing the arithmetical operations indicated by the algebraic symbols. This numerical value will, of course, depend upon the particular values assigned to the letters. Thus the numerical value of 2 a3 is 54 when we make a = 3, for the cube of 3 is 27, and twice 27 is 54. The numerical value of the same expression will be 250 if we make a = 5; for the cube of 5 is 125, and twice 125 is 250. The numerical value of a polynomial undergoes no change, however we may transpose the order of the terms, provided we preserve the proper sign of each. Thus the polynomials 4 a3 — 3 a2 b + 5 a c2, 4 a3 + 5 ac2 − 3 a2 b, 5ac3ab + 4 a3, have all the same numerical value. This follows manifestly from the nature of arithmetical addition and subtraction. XX. Of the different terms which compose a polynomial, some are preceded by the sign, others by the sigu The former are called additive, or positive terms, the latter, subtractive, or negative terms. The first term of a polynomial is not in general preceded by any sign; in that case the sign + is always understood. Terms composed of the same letters, affected with the same exponents, are called similar terms. Thus, 7 ab and 3 a b are similar terms, so are 6 ac and 7 a2c; also, 10 ab3 c'd and 2 a b3 cd; for they are composed of the same letters, and these letters in each are affected with the same exponents. On the other hand, 8 ab3c and 3 a2 b3 c are not similar terms, for although composed of the same letters, these letters are not affected with the same exponents in each. Examples of the numeral values of algebraic expressions:— Let a 4, b = 3, c = 2; then will (1) a + b c=4+32=7—2=5 (2) a2 + ab + b2 = 42 + 4 × 3 + 32 = 16 + 12 + 9 = 37 (5) √(a+b)c−√(a−b) c3 =√(4+3)×2—√√(4—3) × 23 = √√14— √√⁄8 1. ADDITION is the collecting of several similar quantities into one term or sum, and the connecting of dissimilar quantities by their respective signs. The rule of addition may be divided into two cases: (1) When the quantities are similar, and have the same signs. CASE I. 2. When the quantities are similar, and have the same signs. Add the coefficients; affix the letter or letters of the similar terms, aut prefix the common sign + or - .* Thus a+2a+3a+4a+5a=(1+2+3+4+5)a=15a (-a)+(-2a)+(−3a)+(−4a)= −(1+2+3+4)a= −10a (2a+3b)+(4a+5b)=(2a+4a)+(3b-15bj=6a+86. The truth of this rule is evident; for suppose 3a and 5a are to be added together; then by the definition of a coefficient we have Hence — 5a + (-3a) = (−a) + (—a) + (-a) + (-a) + (—a) + (-a) + (−a) + (~) Br )=8a. 3. When the quantities are similar, and have different signs. Collect into one sum the coefficients affected with the sign +, and also those affected with the sign-; to the difference of these sums affix the common literal quantity, and prefix the sign + or —, according as the sum of the+or-coefficients is the greater.* Thus a - 2a +3a-4a+5a= (1 +3 +5) a − (2+4) a=9a—6a=3a And 3x+4y-2x+3y=(3-2) x + (4 + 3) y = x + 7y. The truth of this will be obvious; for to add 5a and -3a together, we have 5a=a+a+a+a+a —3 a = (-a) + (-a) + (-a) Hence 5a + (-3a) = a + a + a + a + a + (-a) + (−a) + (−a). Similarly 2a + (−5 a) = a + a + (−a) + (—a) + (−a) + (-a) + (−a) = + (-a)+(-a) + (-a) +(-a)+( = 3(-a) = — 3 a. a+b+c+de+f · 3(a+b)√x2-y2 + (a−b)√x2+ya a+b+c-d+e+f (a+b) (x2-y2)+ + 3(a−b) (x2+y2)+ a+b-c+d+e+f a-b+c+d+e+f ―a+b+c+de+f 2(a+b) (x2—y2)'s + 4(a−b)√x2+y2 4. Dissimilar quantities can only be collected by writing them in succession, and prefixing to each its respective sign. Thus 9 ay, — 5 c d, and 3 a b, are dissimilar quantities, and their sum is 9 xy + 3 ab-5 cd. In like manner 2 a b, 3 a b2, 4 a b3 are dissimilar quantities, and their sum is 2 a b + 3ab2+4 ab3; which, however, admits of another form of expression, as will be explained in the rule of Division. When several polynomials, containing both similar and dissimilar quantities, are to be collected into one polynomial, the process of addition will be much facilitated by writing all the similar terms under each other in vertical columns. EXAMPLES. (1.) Add together a x + 2 by +cz; √x + √y + √z; 3 y+−2x+ +3z*. 4 cz-3ax-2by; 2ax-4y-22. ax+2by + c z + √ x + √y + √ z -3 ax-2 by + 4 c z − 2 x + 3y++ 3 z+ 2 ax (2.) Add together, 4 ab+3c3 d- 9 m2 n; -4√y-2z+ 5 cz−√x + 2/z = sum required. 4 m2n + a b2 + 5 c3 d + 7 a2 b; 6 m2 n − 5 c3 d +4 m n; -8 a b2; 7 m n2 + 6 c3 d − 5 m2 n — 6 a2 b; 7 c3 d — 10 a b2 - 8 m2 n — 10 d'; and 12 a2b-6 ab2 + 2 c3 d+mn. Arranging the similar terms in vertical columns, we have 4a2b+3c3d — 9 m2 n a b2 + 7c3d - 8 m2n - 10 a b2 12ab+2c3 d 17 ab+18 c3d-12 m2n - 23 a b2 + 11 m n2 — 10 ď + mn = sumn (3.) Add 11 b c +4ad-8ac+5cd; 8 ac+7bc−2 ad+4 mn; 2cd -3ab5ac + an; and 9 a n-2bc-2ad+5cd together. (4.) Add together 2ab+3ac8cx+962x - 8hy-10ky 5a3-4 a b 7bx2- bx 4 ky2-15hy +1463 - 22 a c2 10 x2 19 ac2-8b2x + 9 x2 + 6hy + 2ky2+2ab1 (5.) Add together a3· b3+3 a2b―5 a b2; 3 a3 — 4 a2 b + 3 b3 — 3 ab2; a3 + b3 + 3 a3b; 2 a3 · 4 b3 — 5 a b2; 6 a2 b + 10 a b2, and — 6 a3 — 7 a2 b + 4 a b2 + 2 b3. - (6.) Add‚⁄x2+y2—√/x2 — y2 — 5x y; — 3 (x2 — y2)*+8x y−2 (x2+y2)*; 2√2 + y2 -- 3 xy-5√x-y2; 7xy+10 √x-y2 - 12 √x2+y, and zy + √x2 - y2++ y2 together. ANSWERS. (3.) 16bc5ac + 12 cd + 4 m n-3 a b + 10 an. (4.) 5a3+14 b3-8c x2-7 b x2x2+11x-9hy2-2ky-5ky-9hy. (5.) a3+ab + a b2 + b3. (C.) 2 √x — y3 — 10 √x2 + y2+8x y. 5. When the coefficients are literal instead of numeral, that is, denoted by letters instead of numbers, their sum may be found by the rules for the addition of similar and dissimilar terms; and the sum thus found being enclosed in a parenthesis, and prefixed to the common literal quantity, will express the sum required. |