17. If the product and the multiplier have the same sign, what is the sign of the multiplicand? If they have opposite signs, what is the sign of the multiplicand?

18. If the dividend and the divisor have the same sign, what is the sign of the quotient? If they have opposite signs, what is the sign of the quotient?

19. In the expression a-3m+4p-7b+ n − 15, inclose the third and the fourth terms in a parenthesis preceded by the minus sign, then inclose this parenthesis and the term immediately preceding and the one immediately following it in brackets preceded by the minus sign, leaving the final expression of the same value as the original polynomial.

20. How can you test the correctness of the factors of an algebraic expression?

21. What is the difference in meaning between 3x and x3? Illustrate when x = 5.

22. What is the difference in meaning between the square of the difference of two numbers and the difference of the squares of the same numbers? Illustrate when the numbers are a and b.

23. Why do we change signs when removing a parenthesis preceded by the minus sign?

24. Give the rule for squaring a binomial.

25. When is a binomial the product of the sum and the difference of two numbers?

26. What must be added to x2 + 4x to make it an exact square? What must be added to x2 + 6x + 4 ?

27. Is the product changed if an even number of its factors have their signs changed? Compare the value of (a - b)2 with (ba)2.

28. State the rule for cubing a binomial.

29. Define equation; identical equation; conditional equation.

31. Describe briefly the steps used in solving an equation. What is meant by transposing? What principles are used in transposing?

32. What important difference is there between the equations (x-1)(x+1)= x2 - 1 and 22 – 1 = 0.

33. Subtract 1-x+2x2 from 23. Subtract the same expression from 0.

34. Divide 2-7x+k by x-2, giving quotient and remainder. How long should such divisions be continued? For what value of k will this division be exact?

In examples 35 to 45, Aa2 + 3 ab - 4 b2, B = a3 + 4 a2b - ab2-4 b3, C= a + 4b, D = a3 + 64 b3.

Perform the indicated operations:

39. The minuend is B and the difference is D; find the subtrahend.

40. The divisor is C, the quotient is A, and the remainder is 16 b3; find the dividend.

41. Find the value of B when a = - 2 and b = 3.

42. Find the value of B when a = b.

43. Multiply B by C and verify the result by using a = 1, b = 2.

44. A is quotient, C is divisor; find dividend.

(b) 9 m2 + 16p2 - 24 mp = (3 m – 4p) (?).
(c) 8x-12xy + 6 xу2 — y3 = (? — ?)3.

(d) 9 x1y 36 m2n2 = (3x2y2 - 6mn) (?).

48. Simplify 2 a(a − h)2 — (a2 — 3 ah)a — (a — h) (a — 3 h)a.

49. Simplify (8x3- 12 x2y + 6 xу2—y3) ÷ (2 x—y)2+(y-2x). 50. Divide [3x2 (x + a)2+(x + a)] by (x + a).

51. If you add to a number

What is the number?

Solve the following equations:

52. 3.5 x 9.3 = 1.25 + 10.3.

53. 256(3x-15)= 5.

of it and 7, the result is 27.

54. 33(2x+4)= 6 − 4 (2 x + 3).

55. 2(x-3)-3(1 − 2 x) = 3(2x) — 2(5 – 3x).

56. (x+5)2 - (x2 + 95)= 0.

57. (6x+4) (8 x

5)-(4x+12) (12 x 21)= 0.

58. (x+12) (x − 12) − (x + 8)2 = 0.

59. Using a as the unknown number, write equations whose solutions will answer the following questions:

(a) What number added to 23.7 gives 14.81?

(b) What number subtracted from 12.84 gives 14.81? (c) What number multiplied by 98 gives 12.25?

(d) What number multiplied by gives 12.25?

(e) To what number must be added if the result is to be equal to that obtained by multiplying the number by ?

62. What must be added to x1 − 3 x3 − x + 5 to produce x-1?

63. Solve (4x-1)(x+3)-4 x2 - (- 10 x + 3) + 6 = 0.

64. (a548-17 a3 +52 a + 12 a2)÷(a 2 +a2)=? 65. Find (x+1)3(x-1)3 when x=.

66. Simplify a - [3 a − b − 2(b − a) + 3(a − 2 b)].

67. Divide 2 by x+1.

68. State in algebraic symbols the type forms of multiplication given as special products.

(c) (19xy + 8 x2y2) ÷ (1 − 8 xy).

70. Electric light bills are paid at the rate of 14¢ each for the first few units used and 4¢ each for the remainder. A bill for 35 units was $2. How many units at each price were paid for?

71. Think of a number, double it, add 13, subtract 5, divide by 2. Show that the final result will always be 4 greater than the number you first thought of.

72. Think of a number, multiply it by 3, add 6, divide by 3, subtract the original number. Show that the result will always

be 2.

73. Divide x3 10 x + 17 by x a until the remainder does

not contain x.

Compare the remainder with the dividend.

74. Divide 3 - 5 by x

-a until the remainder does not con

tain x and compare as in example 73.

75. Divide 23 -— 5 by x − 2.

VIII. FACTORING

192. If two or more algebraic expressions are multiplied together, the result is their product, and the expressions multiplied are factors of the product.

..

Thus, 3 x 5 = 15. 3 and 5 are factors of 15, also m(x + y) = mx + my. Here mx + my is the product of which m and (x + y) are the factors.

NOTE. Unless otherwise stated, expressions containing fractions or indicated roots are not considered as factors. Thus, although 3 = 5 × †, or √3 × √3, we shall not in this chapter consider these expressions as factors of 3.

193. Prime Number. A number which has no integral factors except itself and 1 is a prime number.

Thus, 7, 23, a + b, a2 + 3 b2 are prime numbers.

Prime numbers used as factors are prime factors.

Thus, a and a + b are the prime factors of a2 + ab.

194. The student will recall that division is the process of finding one of two factors when their product and the other factor are given. In factoring it is required to find both factors when only the product is given. Thus factoring, like division, is an inverse of multiplication.

In arithmetic we learned a multiplication table and could factor all products that occur in the table from memory. For example, 42 = 6 × 7.

Corresponding to this we have in algebra some type forms of multiplication (Chapter V), and we shall be able to factor the corresponding products from memory.