Arranging the similar terms in vertical columns, we have 4 ab+3c3 d 9 m2 n 17 a2b+18 c3d— 12 m2 n — 23 a b2 + 11 m n2 — 10 d' + m n = sum (3.) Add 11 b c +4ad−8 ac+5cd; 8 ac+7b c−2 ad + 4 mn; 2cd −3 ab +5 ac+an; and 9 a n−2 b c −2 ad+5cd together. 19 a c-8b2x + 9 x2 + 6hy + 2 ky2+ 2 ab3 (5.) Add together a3-b3+3a2b-5ab2; 3 a3-4 a2 b + 3 b3-3 a b2; a3 + b3 +3 a2b; 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 — y3 —5xy; — 3 (x2 — y2)*+8x y−2 (x2+y3)*; 2√x2+ y2 −3xy-5√x2-y; 7xy + 10 √x-y2 — 12 √√x2 + y2, and zy + √x2 −y2 + √x2 + y2 together. ANSWERS. - (3.) 16 b c + 5 ac + 12 c d + 4 m n − 3 a b + 10 an. (4.) 5 a3 + 14 b3 — 8 c x2—7 b x2 — x2+11x—9 hy3 — 2 ky3 — 5 ky —9hy. (5.) a3+ab + a b2 + b3. (6.) 2 √√x2 — y2 — 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. (5) Add a x2+by+c to dx2+hy+k. (6) Add together x2 + xy + y2; a x2-axy+ay2; and—by2+bxy+bx3 (7) Add ↓ (≈ +y) and + (xy). Also + xy + y2 and (x 2 - and x2 — xy+y3. 2 (8) What is the sum of (a+b) x + (cd) y − x √2; (a−b) x + (8c+2d)y+5x√2; 2 bx +3dy -2x√√2; and -3 bx-dy-4x√2. ANSWERS. (3) (a+c)√x-2 (m-n) √y-6√2. (4) gy2 - (2c+ 2 d) x2 + ( a − b + c −d) xy. (5) (a + d) x2 + (b + h) y +c+k. (6) (1 + a + b) x2 (7) First part x. + (1 − a + b) x y + (1 + a − b) y2. (8) (2a-b)x+(4c+3 d) y -2x√2. SUBTRACTION. 6. THE subtraction of monomials is indicated by placing the sign between the quantity to be subtracted and that from which it is to be taken. Thus a-b signifies that the quantity denoted by b is to be subtracted from that denoted by a; and if 2 xy is to be subtracted from x2 + y2, the result is represented by a2 + y2 — 2xy. Place the quantity to be subtracted under that from which it is to be taken; change the signs of all the terms in the lower line from + to and from to +, or else conceive them to be changed, and then proceed as directed in Addition. It is evident, that if all the terms of the quantity to be subtracted are affected with the sign +, we must take away, in succession, all the parts or terms of the quantity to be subtracted; and this is indicated by affecting all its terms with the sign Also, if c-d is to be subtracted from a + b, then c taken from a + b is expressed by a+b-c; but if c-d, which is less than c by the quantity d, be taken from a + b, the former difference, a + b — c, will obviously be too small, and will require the addition of d to make up the deficiency; and therefore c d taken from a + b is expressed by a + b· c+d, which is equivalent to the addition of -c+d to a + b. Hence the reason for the change of the signs in the quantity to be subtracted. Or thus: Since cd is to be subtracted from a +b; then, if c be subtracted, we shall have subtracted too much by d; hence the remainder a + b − c is too small by d; and therefore, to make up the defect, the quantity d must be added. 2a2+ ax + x2-12a3x+20ax3— 4x3 +6a2x2-10ax3 (11) 4y2-4yx+x2-2a(x+y)+6√√/a2—x2—8°/b2—yTM 7. In order to indicate the subtraction of a polynomial, without actually performing the operation, we have simply to enclose the polynomial to be subtracted within brackets or parentheses, and prefix the sign Thus, 2a -Sab+4ab2-(a3+b3+ab3) signifies that the quantity a+b3+ab is to be subtracted from 2a3-3a2b+4ab2. When the operation is actually performed, we have by the rule 2a3-3a3b+4ab3—(a3+b3+ab2)=2a3—3a2b+4ab3—a3—b3—ab3 8. According to this principle, we may make polynomials undergo several transformations, which are of great utility in various algebraic calculations. Thus, a3-3ab+3ab3—b3—a3—(3a2b—3ab2+b3) =a3-b3-(3a3b-Sab2) And x-2xy+y=x2-(2xy-y')=y'-(2xy—x2). (3) From m2 n2x2-2 m np q x+p3 q2 take p2 q2x2-2pqmn x+m2 n2. (6) From (2a-5b)√x+y+(a−b) x y-cz take 3 b x y−(5+c) z— (7) From 2x-y+(y—2x)—(x—2y) take y—2x-(2y—x)+(x+2y). (8) To what is a+b+c~(a—b)—(b—c)—(—b) equal? ANSWERS. (8) (m2 n2-p2 qo) x2+p2 q2-m2 n2 or (m2 n2-p2 q2) x2—(m2 n2—p2 q2). (4) (a—4)(x+y)—(a+2b) xy+(c+7)(x−y). ? (5) 26 (x+y)—2c (x—y)+h2-k2? (6) (5a-6b)x+y+(n−4b) xy+5z2. (7) y-x. (8) 26+2c. 95 MULTIPLICATION. 9. MULTIPLICATION is usually divided into three cases:— (1) When both multiplicand and multiplier are simple quantities. (2) When the multiplicand is a compound, and the multiplier a simple uantity. (3) When both multiplicand and multiplier are compound quantities. CASE I. 10. When both multiplicand and multiplier are simple quantities. To the product of the coefficients affix that of the letters. Thus, to multiply 5 a x by 4 axy, we have 5 × 4 = 20; a x × axy = a2x2y; .. 5ax×4axy = 20 × a2x2y = 20 a2x2 y = product. RULE OF SIGNS IN MULTIPLICATION.* The product of quantities with like signs, is affected with the sign +; the product of quantities with unlike signs, is affected with the sign or + multiplied by + and — multiplied by — give +; or like signs produce + and unlike signs The truth of this may be shown in the following manner:— (1) Let it be required to multiply + a by + b. -; Here a is to be taken as often as there are units in b, and the sum of any number of quantities affected with the sign +, being +, the product ab must be affected with the sign +, and is therefore +ab. (2) Multiply + a by — b, or - a by + b. In the former case -b is to be taken as often as there are units in a, and in the latter a is to be taken as often as there are units in b; but the sum of any number of quantities affected with the sign — is also; hence in either case the product ab must be affected with the sign —, and is therefore ab. • Let N represent either a number or any quantity whatever, and put Now, if in these four last equations we substitute the values of a and b from the first two equations, we have Now, in each of these formulas, the sign of the second number is what is named the product of the two signs of the first number; hence the truth of the rule of signs. |