13. (c2 — c −1)(c2 + c − 1)(c2 − 1). 14. (n2n+1)(n2 + n + 1) (n* — n2 + 1). 16. (2x-5)(3x+1)(2x+5) (3x-1). 17. (cd -3)(cd +7) (2 cd − 1)(3 cd +2). 18. (a2 — 1)(a2 — 5)(a2 + 1)(a2 +5). 19. (9 x2-3x+1)(4x2 + 2x+1)(3x+1)(2 x − 1). 20. (a+b)(a+m). 21. (3 a + x)(2a + y). 22. (am — x)(am — y). 23. (3 cd-m) (2 cd +n). 24. (a+c)(b+d). 25. (3a+2b)(2 c − 5 d). 26. (m3 +2)(m2 — m). 27. (m+n+1)(m − y). ' Perform the indicated operations and simplify : 28. 52(x+1)(2x-3)-3 x (x-1). 29. (a− 2)2+(a + 3)2 −2(a2 + a +4) — 1. 30. (2 m −1)(m+3) − (4 m + 1) (2 m −5) — (1—3 m)(1+2 m). 31. (2a-3)2-3 a(a − 2) — (3 — a)3. 32. cd(cd+1)+cd(cd +1)(cd +2) — c2d2(cd +4). 33. b2(b2+b −1) — b(b2 − b + 1) − b2(b2 — 1). 34. mn(mn-1) — [(mn − 1)2 — (1 — mn)]. 35. (m −2 n − 3)2 + m(2 n + 3 − m) + 2 n(m − 2 n − 3). 36. (1-x)(1 − y) +x(1 − y) +y. 37. a(bm)+b(m − a) +m(a - b). 38. (x+a)(x-a) + (a + z) (a − z) + (x+2)(z − x). 39. a2(b−x)+b2 (x − a) + x2 (a − b ) + (b − x) (x − a) (a − b). - 40. (a+b+c+d)2 - (a-b-c-d)2. 41. (a +x+1)2 + 2 (a + x + 1)(a + x − 1) + (a + x − 1)2. 42. (a+1)3 — 3 (a + 1)2(a − 1) +3 ( a + 1)(a − 1)2 — (a — 1)3. SOM. EL. ALG. — 4 72. Multiplication with Literal Exponents. The literal exponent is constantly used in the later discussions of algebra, and familiarity with this form is readily attained in the processes of multiplication and division. Illustrations: 3m 1. Multiply a3m + a2m −2 am +3 by a2m+am — 1. -3x2n + 6 x2n-1- 9 x2n-2+3x2n−8 x2n+1-5 x2n +9x2n-1 — 10 x2n-2 + 3 x2n-8 Result. n-1 -C. a2m by a2m+1. 12. x2+1 - 2 x1 − 3 x2-1 +4 x2-2 by x2 -3x2-1. 13. a3m — a2mb1 + amb2n — b3n by a2m +ambn + b2n. 14. 22m+3n+1 + x2m+2n + 3 x3m+n−1 + x1m-2 2m-2 +x5m-n-s by xm+n-1 + x2m−2 + x3m-n-3 ̧ 73. Multiplication with Detached Coefficients. In many cases the labor of multiplication and division is lessened by the use of detached coefficients. Since the product of two expressions is an expression whose degree is the sum of the degrees of the given expressions (Art. 67), we supply the necessary x-factors for the coefficients Missing Powers. If any power of a literal factor is missing, its coefficient is 0, and the term must be provided for in the sequence of powers by an inserted 0. Thus, (x8-2x2+3) (x3+x−1) = (x3 − 2 x2 +0x+3) (x2 + 0 x2 + x−1). Multiplying with the coefficients detached, For practice in multiplication with detached coefficients, use Ex. 9, 13 to 24 inc. At the discretion of the teacher the foregoing paragraph and the corresponding operation suggested under Division may be omitted on the first reading of the text. For review topics, however, the methods have a distinct value. CHAPTER V DIVISION. REVIEW 74. Division is the process of finding one of two factors when their product and one of the factors are given. The dividend is the given product; the divisor, the given factor; and the quotient, the factor to be found. THE NUMBER PRINCIPLE OF DIVISION 75. The Law of Distribution. The quotient of a polynomial by a monomial equals the sum of the quotients obtained by dividing each term of the polynomial by the monomial. The significance of this law is that the divisor is distributed as a divisor of each term of the dividend. The process of division being the inverse of the process of multiplication, the laws governing signs, coefficients, and exponents in multiplication form, when inverted, the corresponding laws for division. Therefore, from (1) and (4), and from (2) and (3): 76. Like signs in division give a positive result. 77. Unlike signs in division give a negative result. COEFFICIENTS IN DIVISION 78. The coefficient of a quotient of two algebraic expressions is obtained by dividing the coefficient of the dividend by the coefficient of the divisor. In general, therefore, we have the following: If m and n are any positive integers and m is greater than n amax a × a to m factors, ... Hence, the statement of the second index law is made as follows: 79. The quotient of two powers of a given factor is a power whose exponent is the exponent of the dividend minus the exponent of the divisor. |