57. Multiplication of Polynomials by Monomials.-The third Multiply each term of the multiplicand by the multiplier, and add their products, observing the law of multiplication that like signs give plus and unlike signs give minus. EXAMPLE.-Multiply 2.3-4x+5 by 9 x3. By the rule, (2.3-4x+5)X(-9.c3)=(2.r3) (-9.c3)+(-4.c) (-9.x3)+(5) (—9.x3) 18.36.x1-45.c3. == 58. Multiplication of Polynomials by Polynomials. From the preceding sections, (a + b) (c — d) = (a + b)c + (a + b) ( − d) ac+be (a+b)d [Law V, and 41, 6] 1. (a + b) (c + d) = (a + b) c + (a + b) d [Law V, 87] 2. [Law V, 87] = 41,6] = [Law V, 7] [241, 6] [Law V] = ac+(—b)c+a(−d)+(-—b)(—d) [Law V] = ac bead + bd. Whence follows from equations 1, 2, 3, 4, the rule: [241, 6, 8] Multiply each term of the multiplicand by each term of the multiplier; if the terms have the same sign, prefix the sign + to their product; if they have different signs, prefix the sign; then add these partial products to form the complete product. 1. Multiply 2a-36 by 3a-7b. According to the rule, multiply 2a-3b by 3a and then by --7b and add the partial products. In practice the work is usually simplified by arranging similar terms in the same column. 2 a Thus, 2. Multiply 61+6x-2-3x3-x by 2x3 + 2 + x. It is convenient to arrange the multiplicand and multiplier in the same order of powers of some common letter and to write the partial products in the same order. A polynomial is arranged in descending powers of some letter, x, if the highest power of a comes in the first term, the next highest in the second term, and so on; in ascending powers of x, if the powers of x are arranged in the reverse order. Arrange the multiplicand and multiplier with respect to the descending powers of x. For example, 59. The rule in 154 has an application which is very useful in consequence of the different forms in which the product of several binomial or polynomial factors may be written. Thus, (a — b) (d — c). 60. In like manner it can be shown that in the indicated product of more than two expressions, the signs of any even number of them may be changed without altering the product, but if the signs of any odd number of them are changed, the sign of the product is changed (256). Thus, (a - b) (cd) (e-f) may be written in any of the forms (ab) (dc) (ƒ— e) (ba) (d-c) (e-f) (ba) (cd) (ƒ—e), etc. 9. 10. Multiply: (8.xyz3) (—9.x2yz2) (— 5.x3yz) = (7 am2n) (— 3 b2n2) (— 4 ab) (ab2n2) (— 2 b2n2m) (— m2n) = 25. (8x-7y) (7x+6 y). 29. (7.25 +4 x) (2.8 — 3.6 x). 31. (7a-0.3) (2.8 a +5}). 33. (a+b-c) (a-b+c). 35. (3a+b-x)2. 37. (3x-5y-2). 39. (aa+2ab+bb) (a - b). 41. (xxxy+yy) (x + y). 43. 44. 45. 46. 47. 48. 49. 50. 34. (a+b+c)2. 36. (2a-3b+x)2. 38. (aa-2ab+bb) (a+b). 42. (xxxyyy) (xxxy+yy). ((a+b)+(x + y)) ((a+b) — (x+y)). (a − b + c − d) (a+b— c — d). (3a+2b+5x-y) (3a+2b-5x+y). (6 ac-3 ad+2 bcbd) (6 ac-3 ad+2bc+bd). (4 ab-6 ax2 by -3xy) (4 ab+6ax-2 by-3 xy). 51. (a+b) (a+b) (a+b). 53. (x+1)3. 52. (ab) (a — b) (a — b). 54. (1―y)3. CHAPTER V POSITIVE AND NEGATIVE NUMBERS DIVISION 61. Numerical Division.-Division is the operation inverse to multiplication. To divide a by b, is to find the number which multiplied by b produces a. The result is called the quotient of a by b, and is written. Hence, in accordance with the definition of division, if follows formally, As in the case of subtraction, division can not always be accomplished. It is only in a special case that a group of a things can be regrouped into subgroups each containing consider the following a and b groups. individuals. For The a group or 12 group. The b group or 4 group. Here the 4 group can be counted out of the 12 group three distinct times; that is, the quotient of 12 by 4 is 3. But if the a group contained 15 things and the b group 4 things, then after counting out 3 groups of 4 things each from the a group, there would not remain enough things to form a fourth group of 4 things; and accordingly 4 is not contained an exact number of times in 15. The number a to be divided is called the dividend, the number divided by is the divisor, and the result of the division is the quotient. Formula IX translated, is: The product of the quotient by the divisor is equal to the dividend. The number of times a group of 4 individuals can be counted out of another group of 4 things is 1, or a group of a things out of another group of a things is 1. Thus: which satisfy the definition of division: (quotient, 1) × (divisor, 4) = dividend, 4 and 62. Numerical Division Gives a Single Result.-When division can be effected at all, it leads to but a single result; it is determinate. There is but one number whose product by b is a. a quotient has two values, c and d; then by IX, 261, Suppose the с because b groups of c individuals each can not be equal to 6 groups of d individuals each unless c d (24). NOTE. The case b =0 is excluded, since 0 is not a number in the sense in which that word is used in this discussion. This theorem is of vital importance. It declares that if a product and one of its factors are determinate the other is also determinate; or should one of the factors of the product change while the other remains unchanged the product changes. The possibility of division in the arithmetical sense depends upon the truth of this theorem alone. The fact that Law XI does not hold for 0, that we can not divide by 0, is clear; for if one of the factors of a product is 0, the product is 0, however the other factor may change. Thus let the quotient of beq; then by definition of division q × 0 should equal 5; but q×00 so long as q is any known fixed quantity whatever, hence the assumption that can have a definite fixed value is false. 5 |