-3ab4ab2-(a3+b3+ab3) signifies that the quantity a3+b3+ab is to be subtracted from 2a3—3a2b+4ab2. When the operation is actually performed, we have by the rule, 2a3-3ab4ab3—(a3+b3+ab2)=2a3-3a2b+4ab2—a3—b3—ab2 When, therefore, brackets are removed which have the sign the signs of all the terms within the brackets must be changed. before them, 8. According to this principle, we may make polynomials undergo several transformations, which are of great utility in various algebraic calculations. Thus, And a3-3ab+3ab2 — b3 — a3 — (3a2b—3ab2+b3) a3-b3-(3ab-3ab2) =a33ab2-(3a2b+b3) =-(—a3+3a2b—3ab2+b3) x2 —2xy+y2= x2 — (2xy—y2)=y2 — (Qxy —x2). (3) From m2n2x2-2mnpqx+p2q take p2qx2-2pqmnx+m2n2. (4) From a(x+y)-bxy+c(x-y) take 4(x+y)+(a+b)xy-7(x—y). (5) From (a+b)(x+y)—(c—d) (x−y)+h2 take (a−b)(x+y)+(c+d) (x−y)+k2. (6) From (2a-5b)√x+y+(a−b)ry-cz take 3bry-(5+c)22-(3a-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? (9) From A2+Bx2+Cx+D take A123+B1x2+C1x+D1• ANSWERS. (3) (m2n2-p2q3)x2+p2q2-m2n2, or (m2n2-p2q2)x2-(m2n2-p2q2), or (m2n3 -p2q) (x2-1). (4) (a-4) (x+y)=(a+2b)xy+(c+7) (x−y). (5) 2b(x+y)—2c(x—y)+h2 —k2. (6) (5a−6b)√x+y+(a−4b)xy+5z2. (7) y+x. (8) 2b+2c. (9) (A—A1)23+(B−B1)x2+(C—C1)x+D—D ̧. 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 quantity. (3) When both multiplicand and multiplier are compound quantities. CASE I. 10. When both multiplicand and multiplier are simple quantities, or monomials. To the product of the coefficients affix that of the letters.* Thus, to multiply 5r by 4y, we have 5×4=20; x×y=xy; •*.5x × 4y=20 ×xy=20xy= product. 11. Powers of the same quantity are multiplied by simply adding their indices; for since, by the definition of a power, a5- ...aaaaaaa×aaaaaaaaaaaaaaaaaaaa12=a5+7. .... Also, aaaa.... to m factors; a"aaa. to n factors; ..aTM×a"=aaa.... to m factors Xaaa.... to n factors; -аааааа..... to (m+n) factors; It is proved, in the same manner, that aTM×aa× ah × ak=am+n+b+k ̧ *I. The rule is derived in the following manner: We begin by assuming that when several letters are written one after another without any sign, their continued multiplication is understood, and that the operation proceeds from left to right. Then abcd will sig nify a multiplied by b, that product by c, and that again by d. We shall now prove that in whatever order these letters or simple factors are arranged, their continued product will always be the same; and, moreover, that they may be grouped into partial products at pleasure, provided all the letters be employed each time. Thus the above product may be written bade (the multiplication here, as before, going on by each factor successively from left to right), and the result will be the same as before; or it may be written aXbXcd, understanding the products separated by the sign X as being previously formed and then multiplied together. The demonstration depends upon three propositions, which we shall first establish: (1)...aXb=Xa b 1 1 1 1 1 1 1 a 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 For in the adjoining table of units let b denote the number of units in each horizontal row, and a the number of rows, then b multiplied by a, or repeated a times, will give the number of units in the table. But a, which is the number of horizontal rows, is also the number of units in each column; and b is the number of columns; then a multiplied by b, or repeated b times, will produce the number of units in the table again; whence b multiplied by a is equal to a multiplied by b. In a similar manner, from the adjoining table, it may be proved that II. By (1) abcd=bacd=by (2) bcad=by (2) bcda. Thus, we perceive that the factor a has been made to occupy successively every place from the first to the last. The same might now be done with the factor b, and so with all the others. Therefore a product is the same, whatever be the order of its factors. III. Again. Take aXbXcxdxe. It may be written by (3) aXbcXdXe or by (3) abcde, or, instead, by (3) abXcd Xe. From which it appears that the factors of a product may be grouped into partial products at pleasure, and then afterward multiplied together or conversely. IV. Let us now suppose that the product 3a3b2 is to be multiplied by the product 5ab4, Instead of multiplying by the whole product 5ab4, multiply by its factors separately, and we have 5a2b+3a3b2. Since the order may be changed at pleasure, bring the numerical factors together, and the different powers of the same letters; thus, 5×3a2a3b+b2. Grouping the different powers of the same letters into partial products, as well as the numerical factors. the result is 15a5b7, which has evidently been obtained by multiplying the coefficients and adding the exponents of like letters. † Such a relation as that of a product to its factors is called a symmetrical relation. 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 The continued product of an even number of negative factors is positive; of an uneven number, negative.* 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.† (7) Multiply a+b+√x2—y2-3ry by -2√x. (8) Multiply a"+by"—c"y"—d Let m, m' be two monomial quantities whose product is required. If m, m' are both additive quantities, the product mm' is an additive quantity. This is the case of arithmetic. If the multiplicand m is an additive quantity, and the multiplier m' a subtractive quantity, the expression m×(—m') indicates that the multiplicand m is to be subtracted as many times as there are units in m', or that m' repetitions of the quantity m are to be subtracted, which is expressed by ―mm'. If m is subtractive and m' additive, -m taken once is -m; taken twice is - -2m; en m' times is -m'm. tak If m and m' are both subtractive, the quantity -m is to be subtracted m' times. Now -m subtracted once +m, twice is +2m; and m' times is +m'm. 1st. Suppose the signs to be all plus. The whole multiplicand being to be taken as many times as is denoted by the multiplier, each of its parts or terms must be taken so many times. 2d. For the case where some of the signs are negative, see the demonstration in the next note. CASE III. 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 prod uct.* (6) Multiply 4a3—5a2b—8ab2+2b3 by 2a2—3ab—4b2. 8a5—10a1b-16a3b2+ 4a2b3 -12ab+15a3b2+24a2b3- 6ab -16ab20ab3+32ab1-8b5 8a3-22a1b-17 a3b2+48a2b3+26ab*—8b5= product. a-b C -d ac-be * 1st. Suppose all the terms of the multiplier to be affected with the sign +. The mul tiplicand, being to be taken as many times additively as is denoted by the multiplier, must be taken as many times as is denoted by each term of the multiplier separately, and the separate results added together. 2d. When there are both additive and subtractive terms in the multiplier and multiplicand. The rule for the signs may be thus demonstrated. Let a-b be multiplied by c-d. First multiplying a by c, the product is ac; but b should have been subtracted from a before the multiplication; bunits have, therefore, been taken c times in the a, which ought not to have been so taken; hence b, taken c times, must be subtracted, and there results ac-be as the product of a-b by c. But the multiplier was c-d instead of c; therefore the multiplicand has been taken d times too often; d times the multiplicand, which will be of the same form as c times the multiplicand, viz., ad—bd, must be subtracted, and the rule for subtraction is to change the signs of the quantity to be subtracted. The result is, therefore, ac-bc-ad+bd; comparing which with the given quantities we perceive that like signs have produced + and unlike To render the demonstration still more general, a may represent the assemblage of the additive terms of the multiplicand, and b that of the subtractive; c and d the same for the multiplier. ad-bd ac-be-ad-bd. The results in examples (1), (2), and (3) show, 1. That the square of the sum of two numbers or quantities is equal to the square of the first of the two quantities plus twice the product of the first and second, plus the square of the second. 2. That the product of the sum and difference is equal to the difference of the squares; and, 3. That the square of the difference is equal to the sum of the squares minus twice the product. (7) Multiply a'b-ab' by h'k-hk'. a'b-ab' h'k-hk a'bh'k-ab'h'k —a'bhk'+ab'hk' a'bh'k-ab'h'k-a'bhk' +ab'hk' = product. (8) Multiply +xTM-1y+xTM-2y2+xm¬3y3+ &c., by x+y. xm+xm1y+xm¬2y2+xm¬3y3+..... (9) Multiply x2+y2 by x2—y2. (10) Multiply x2+2xy+y2 by x—y. (11) Multiply 5a*—2a3b+4a2b2 by a3—4a2b+2b3. (14) Multiply a2+2ab+b2 by a2-2ab+b2. (15) Multiply x2+xy+y2 by x2—xy+y2. (16) Multiply x2+ y2+z2 —xy-xz-yz by x+y+z. (18) Multiply together g+h, g+h, g-h, and g-h. (19) Multiply together p+q, p+2q, p+3q, and p+4q. (22) (5a-4b1y°) × (5a3x3+4b1y) as ex. 2. (9) x^—y1. (10) x3+x2y-xy2 —y3. ANSWERS. (11) 5a7-22ab+12ab2 —6a1b3 —4a3b1+8a2b3. (12) -2x+1. (13) 5x+ax3-107a2x2+{a3x+fa*. (14) a1-2a2b2+b*. (15) x+xy+y'. (16)+y+3-3xyz. (17) x3-(a+b+c)x2+(ab+ac+bc)x-abc. (18) g*—2gh2+h*. (19) p+10p3q+35pq+50pq+24q. (20) z+-24z3+20622-744z+945. (21) am-am+n+am+_am+1+a+1—a3. (22) 25a06-16b810. When the multiplicand and multiplier are each homogeneous, the product will be also; and the degree of each term of the product will be equal to the sum of the degrees of a term in the multiplier, and a term in the multiplicand. This serves conveniently to verify the accuracy of the operation. It is applicable in the above examples to all except the 12th, 20th, 21st, and 22d. C |