NOTE. Examples involving literal, fractional, and negative exponents will be found in the chapter on the Theory of Indices. 51. Products Written by Inspection. Although the result of multiplying together two binomial factors, such as x+8 and x -7, can always be obtained by the methods already explained, it is of the utmost importance that the student should learn to write the product rapidly by inspection. This is done by observing in what way the coefficients of the terms in the product arise, and noticing that they result from the combination of the numerical coefficients in the two binomials which are multiplied together; thus (x+8)(x+7)= x2 + 8x+7x+56 In each of these results we notice that: 1. The product consists of three terms. 2. The first term is the product of the first terms of the two binomial expressions. 3. The third term is the product of the second terms of the two binomial expressions. 4. The middle term has for its coefficient the sum of the numerical quantities (taken with their proper signs) in the second terms of the two binomial expressions. The intermediate step in the work may be omitted, and the products written at once, as in the following examples: (+2) (æ +3)= +5 +6. (x − 4 y) (x − 10 y) = x2 - 14 xy + 40 y2. (x-6y)(x+4y)= x2-2xy - 24 y2. By an easy extension of these principles we may write the product of any two binomials. Thus (2x+3y) (x − y) = 2x2 + 3 xy − 2 xy — 3 y2 =2x2+xy-3y2. (3 x −4y) (2 x + y) = 6 x2 − 8 xy + 3 xy — 4 y2 3. (x-3)(x + 10). --- 17. (a 11)(a + 11). 13. (a-8) (a + 4). 27. (3x+7)(2x-3). 14. (a 8) (a+8). 28. (4x-3) (2x+3). 52. In the following cases we lessen the labor of multiplication by using the Method of Detached Coefficients: (i.) When two compound expressions contain but one letter. (ii.) When two compound expressions are homogeneous and contain but two letters. Ex. 1. Multiply 2 x3 - 4x2 + 5x-5 by 3x2 + 4x − 2. Writing coefficients only, 2- 4+ 5 5 3+42 Inserting the literal factors according to the law of their formation, which is readily seen, we have for the complete product, 625-4x4-5 x3 + 13x2 - 30x + 10. Ex. 2. Multiply 3 aa + 2 a3b + 4 ab3 + 2 b4 by 2 a2 — b2. 3+2+0+4+2 2+0-1 6+4+0+8+4 -3-2 6+4 4-2 4 3+6+4 In the first expression the term containing a262 is missing, so we write a zero in the corresponding term in the line of coefficients. In the second expression we write a zero for the coefficient of the missing term ab. The law of formation of literal factors is readily seen, and we have for the complete product, 6a6+ 4 a5b - 3 a4b2 + 6 a3b3 + 4 a2b1 — 4 ab5 — 2 bo. EXAMPLES IV. g. 1. Multiply 25 + x2 + x2 + 2 x + 1 by x3 + x − 2. 2. Multiply a3 + 6 a2b + 12 ab2 + 8 b3 by 3 a3 + 2 b3. 3. Multiply 2 at 3a2 + 4 a + 4 by 2 a2 3 a 2. - 4. Multiply 3x5 + 2 x1y − x3y2 + xy1 by x2 + 4 xy − 5 y2. CHAPTER V. DIVISION. 53. When a quantity a is divided by the quantity b, the quotient is defined to be that which when multiplied by b produces a. The operation is denoted by a÷b, % or a/b; in each of these modes of expression a is called the dividend, and b the divisor. Division is thus the inverse of multiplication, and 55. Since Division is the inverse of Multiplication, it follows that the Laws of Commutation, Association, and Distribution, which have been established for Multiplication, hold for Division. DIVISION OF SIMPLE EXPRESSIONS. 56. The method is shown in the following examples: Ex. 1. Since the product of 4 and x is 4 x, it follows that when 4 x is divided by x the quotient is 4, or otherwise, 4x÷x=4. We see, in each case, that the index of any letter in the quotient is the difference of the indices of that letter in the dividend and divisor. This is called the Index Law for Division. We can now state the complete rule: Rule. The index of each letter in the quotient is obtained by subtracting the index of that letter in the divisor from that in the dividend. To the result so obtained prefix with its proper sign the quotient of the coefficient of the dividend by that of the divisor. Ex. 4. Divide 45 a6b2x4 by - 9 a3bx2. The quotient =(− 5) × a6-3b2 -1x4-2 == 5 a3bx2. Ex. 5. 21 a2b3 ÷ (— 7 a2b2) = 3 b. NOTE. If we apply the rule to divide any power of a letter by the same power of the letter, we are led to a curious conclusion. This result will appear somewhat strange to the beginner, but its full significance will be explained in the chapter on the Theory of Indices. |