23. From 3a+b+c-d-10 take c+2a-d. Ans. ab 10. 24. From 3a+b+c-d-10 take 6-19+3a. Ans. c-d+ 9. 25. From a3+ 3b2c + ab2 — abc take b3+ab2 — abc. Ans. a3+3b2c — b3. 26. From 12x+6a-4b+40 take 4b-3a+4x+6d-10. Ans. 8x+9a-8b-6d+50. 27. From 2x−3a+4b+6c-50 take 9a+x+6b-6c-40. - 12a 2b+12c - 10. Ans. x 28. From 6a-4b-12c+12x take 2x-8a+ 4b-6c. Ans. 14a-8b-6c+10x. 38. In algebra the term "difference" does not always, as in arithmetic, denote a number less than the minuend: for, if from a we subtract - b, the remainder will be a+b; and this is numerically greater than a. We distinguish between the two cases by calling this result the algebraic difference. 39. When a polynomial is to be subtracted from an algebraic expression, we inclose it in a parenthesis, place the minus sign before it, and then write it after the minuend. Thus, the expression 6a2 - (3 ab — 2b3 +2bc) indicates that the polynomial 3 ab-263+2bc is to be taken from 6a2. Performing the indicated operations by the rule for subtraction, we have the equivalent expression 6a3ab2b-2bc. The last expression may be changed to the former by changing the signs of the last three terms, inclosing them in a parenthesis, and prefixing the sign. Thus, 6a2-3ab2b-2bc6a2 - (3 ab-2b+2bc). In like manner any polynomial may be transformed, as indicated below. 7a3-8a2b-4b2c+6b3 7 a3 (8ab4b2c-6b3) NOTE. = =7a-8a2b- (4 b2c — 6b3). — 8a3-7b2+c-d=8a3- (7 b2 — c + d) =8a3-762-(c+d). 9b3a3a2-d9b3 (a-3a2 + d) =9b3-a-(-3a2 + d). -The sign of every term is changed when it is placed within a parenthesis which has the minus sign before it, and also when it is brought out of such parenthesis. 40. From the preceding principles we have The sign immediately preceding b is called the sign of the quantity; the sign preceding the parenthesis is called the sign of operation; and the sign resulting from the combination of the signs is called the essential sign. When the sign of operation is different from the sign of the quantity, the essential sign will be; when the sign of operation is the same as the sign of the quantity, the essential sign will be +. MULTIPLICATION. 41. Multiplication is the operation of finding the product of two quantities. The multiplicand is the quantity to be multiplied; the multiplier is that by which it is multiplied; and the product is the result. The multiplier and multiplicand are called factors of the product. Exercises. 1. If a man earns a dollars in 1 day, how much will he earn in 6 days? In 6 days he will earn six times as much as in 1 day. If he earns a dollars in 1 day, in 6 days he will earn 6 a dollars. 2. If 1 hat costs d dollars, what will 9 hats cost? Ans. 9d dollars. 3. If 1 yard of cloth costs c dollars, what will 10 yards cost? Ans. 10c dollars. 4. If 1 cravat costs b cents, what will 40 cost? Ans. 40b cents. 5. If 1 pair of gloves costs b cents, what will a pairs cost? If 1 pair of gloves costs b cents, a pairs will cost as many times b cents as there are units in a; that is, b taken a times, or ab, which denotes the product of b by a or of a by b. 6. If a man's income is 3 a dollars a week, how much will he receive in 46 weeks? NOTE. It is proved in arithmetic (Davies' "Standard Arithmetic," 50) that the product is not altered by changing the arrangement of the factors; that is, 12ab a × b × 12 = b × a × 12 = a × 12 × b. = 42. To find the Product of Two Positive Monomials. Multiply 3 ab by 2a2b. We write, 3 a2b2 × 2a2b= 3 × 2 × a2 × a2 × b2 × b= 3 × 2 aaaabbb; in which a is a factor 4 times, and b a factor 3 times: hence ( 14) 3a2b2 × 2 a2b 3 × 2 a1b3 = 6 a1b3, in which we multiply the coefficients together, and add the exponents of the like letters. The product of any two positive monomials may be found in like manner. Hence the rule: — Multiply the coefficients together, for a new coefficient. Write after this coefficient all the letters in both monomials, 22. 75axyz by 5 abcdx'y'. 23. 64 a3m3x*yz by 8 ab2c3. 24. 9a'b'c'd by 12a3b*c. 25. 216 ab'c'd by 3a3b2c3. 26. 70a b'c'd fx by 12a'b'cdx'y3. 43. Multiplication of Polynomials. (1) Multiply a-b by c. Ans. 375 abcdxyz. Ans. 512abcm3x*yz. Ans. 108 abcd3. Ans. 648 a b°c3d®. Ans. 840abcd3fx3y3. It is required to take the difference between a and b, c times; or to take c, ab times. As we cannot subtract b from a, we begin by taking a, c times, which is ac; but this product is too large by b taken c times, which is be: hence the true product is ac-bc. If a, b, and c denote numbers, as a = 8, b = 3, and c=7, the operation may be written in figures. (2) Multiply ab by c-d. It is required to take a − b as many times as there are units in c d. If we take a b c times, we have ac-bc; ac but this product is too large by a -b taken d times. a-b taken d times is ad- db. Subtracting this product from the preceding by changing the signs of its terms (§ 37, rule), we have (a - b) + (a — c) = ab — bc — ad + bd. 7 -bc 21 35 ad+bd = bc ad + bd From the preceding examples we deduce the following rule: When the factors have like signs, the sign of their product will be +. When the factors have unlike signs, the sign of their product will be →. |