7. In order to indicate the subtraction of a polynomial, without actually per forming the operation, we have simply to enclose the polynomial to be subtracted within brackets or parentheses, and prefix the sign · Thus, 2a -3a3b+4ab2—(a3+b3+ab3) signifies that the quantity a+b+ab is to be subtracted from 2a3-3a2b+4ab2. When the operation is actually performed, we have by the rule 2a3-3ab+4ab2—(a3+b3+ab3)=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, a3-3a2b+3ab3—b3—a3—(3a2b—3ab3+b3) -a-b3-(3a2b-3ab2) =a3+3ab2-(3ab+b3) =-(—a3+3a2b—3a2b+b3) And x-2xy+y=x2-(2xy-y2)=y'—(2xy—x3). (3) From m2 n2x2-2 m np qx+p2 q' take p2 q2x2-2pq m n x + m2 n2. (4) From a (x+y)—bxy+c (x—y) take 4(x+y)+(a+b) x y—7 (x—y). (5) From (a+b)(x+y)—(c−d) (x—y)+h3 take (a—b) (x+y)+(c+d) (x—y)+k2. (6) From (2a-5b)√x+y+(a−b) x y-cz2 take 3 b x y−(5+c) 2'— (3 a—b) (x+y)+. (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. (3) (m2 n2—p2 q2) x2+p2 q2—m2 n2 or (m2 n2-p2 q2) x2-(m2 n2—p2 qo). (4) (a-4)(x+y)—(a+2b) x y +(c+7) (x−y). (5) 2b (x+y)—2c (x—y) +h2—k2. (6) (5a—6b)√x+y+(n−4b) xy+5z2. (7) y---2'. (8) 2b+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 simps 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 az by 4azy, we have 5 × 4 = 20; ar × ar y = a2x2y; .. 5axx 4ary = 20 × a2x2y = 20 a2x2y = 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 —; multiplied by + and +multiplied by and or or multiplied by - give +; 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 a b must be affected with the sign+, and is therefore + a b. (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 — a b. • Let N represent either a number or any quantity whatever, and put N; N Turs, since a m + a, and à m + §, we shall have + = + N + − N N − 1 − + N. Now if in these four last equations we substitute the values of a and b from the first two eqna ✦ (+ N) = + N; + (−− N) = − N − ( + N) = − N; − (− N) = + N. each of these formulas, the sign of the second number is what is named the product of th● tes ngas of the first number; hence the truth of the rule of signs. (3) Multiply - - a by-b. Since by the last case + a multiplied by -b produces a b; and since - a multiplied by-b cannot produce the same product as + a multiplied by b, it is evident that the product of a and b can only be + a b. 11. Powers of the same quantity are multiplied by simply adding their indices; for since by the definition of a power απ =aaaaa; a2 = aaaaaaa .. a3 × a2 = aaaaa × aaaaaaaaaaaaaaaaaaa = a22 Also a aaa.... to m factors; a" = aaa to n factors .. aTM × a" = aaa.... to m factors X aaa.... to n factors .... It is proved in the same manner that aTM×a×a×ak = a+a+b+k 12. When the multiplicand is a compound, and the multiplier a simple quantity. Multiply each term of the multiplicand by the multiplier, beginning at the left hand; and these partial products being connected by their respective signs, will give the complete product. EXAMPLES. (1.) Multiply a2 + ab + b2 By 4 a Product 4a3 + 4a3b + 4 a b3. (2.) Multiply a2 —2ab+b2 Product 3a2xy-6 ab xy + 3 b3xy. (3.) Multiply 5 m n + 3 m2 — 2 n2 by 12 abn. (7.) Multiply a + b + √√x2 - y2 −3 xy by -2x. CASE II. 13. When both multiplicand and multiplier are compound quantities. Multiply each term of the multiplicand, in succession, by each term of the multiplier, and the sum of these partial products will give the complete product () Multiply 4a-5a2b-8ab2+2b3 by 2a2-3ab-46o. 8a-22a1b-17a3b2+48a2b3+26ab1—8b3 = product. (7) Multiply a'b-ab' by h'k-hk'. a'b-ab' h'h-hk a'bh'k-ab'h'k -a'bhk'+ab'hk' a'bh'k-ab'h'k—a'bhk'+ab'hk' = [.roduct. (8) Multiply am+xm1y+xm2y2+233+, &c., by z+9. 2m+xm1y+xm2y2+xTM3+y3+... (9) Multiply x2+y2 by x2—y2. (10) Multiply x2+2xy+y2 by x—y. (11) Multiply 5a1—2a3b+4a2b2 by a3-4a2b+2b3. · (16) Multiply x2+y2+z2—xy—xz-yz by x+y+8. ANSWERS. (9) x1—y1. (10) x3+x2y—xy2—y3. (11) 5a7-22ab+12a3b2—6a1b3—4a3b1+8a2b3. (12) x®—2x3+1. (13) 5x1+žax3-107a2x2 + fa3x+Ja*. (14) a 2a2b2+b1. (15) x1+x2y2+y^. (16) x3+y3+z3-3xyz. (17) x3—(a+b+c)x2+(ab+ac+bc)x—abc. MULTIPLICATION BY DETACHED COEFFICIENTS. 14. In many cases the powers of the quantity or quantities in the multiplication of polynomials may be omitted, and the operation performed by the coefficients alone; for the same powers occupy the same vertical columns, when the polynomials are arranged according to the successive powers of the letters; and these successive powers, generally increasing or decreasing by a common difference, are readily supplied in the final product. EXAMPLES. (1.) Multiply +x2y+xy2+y3 by x-y. Coefficients of multiplicand 1+I+i+] multiplier 1+1+1+1 1+0+0+0—1 |