6. Multiply a2 + 4ax + 4x2 by a3 – 4ɑx + 4x2. 7. Multiply a2 - 2ax + bx-x2 by b+x.

8. Multiply 15x2+18ax - 14a by 4x2 - 2ax – a2. 9. Multiply 2x3 + 4x2 + 8x + 16 by 3x-6. 10. Multiply 2x2 - 8xy + 9y3 by 2x-3y. 11. Multiply 4x3 – 3xy – y3 by 3x-2y.

12. Multiply 3 — xˆy + xy* — y3 by x+y.

13. Multiply x + 2y - 3z by x-2y+3z.

14. Multiply 2x2 + 3xy + 4y by 3x2 - 4xy + y2.

15. Multiply x2+ xy + y2 by x2 + xz + z3.

16. Multiply a2 + b2 + c2 − bc − ca ·

x2

ab by a+b+c.

17. Multiply 3 − xy + y2 + x + y + 1 by x+y-1. 18. Multiply +4x+5x-24 by x-4x+11. 19. Multiply x-4x+11x-24 by x2+4x+5.

20. Multiply 2×3 – 2x2 + 3x − 4 by 4x3 + 3x2 + 2x + 1.

21. Multiply * + 2x3 + x2 - 4x-11 by x-2x+3.
22. Multiply x3 − 5x1 +13×3 – x2 − x + 2 by x3 – 2x – 2.

23. Multiply a1 – 2a3 + 3a2 – 2a + 1

by a* + 2a3 + 3a2 + 2a + 1.

26. Multiply together x-x+1, 2+x+1, and x-x+1. 27. Multiply x* — ax2 + bx2 - cx+d by x+ax3- bx2 + cx — d. Shew that (x+a)* = x* + 4x3a + 6x2a3 + 4xa3 + a*.

28.

29. Shew that x(x + 1)(x + 2) (x + 3) + 1 = (x2 + 3x + 1)2.

30. Multiply together a +x, b+x, and c+x.

31. Multiply together x-a, x-b, x-c, and x-d.

32. Multiply together a+b-c, a + c-b, b+c-a, and a+b+c.

33. Simplify (a + b) (b + c) − (c + d) (d + a) − (a + c) (b–d).

(d+a) −

34. Simplify (a + b + c + d)2 + (a−b-c+ d)2 + (a−b+c-d)3 +(a+b-c-d).

36. Simplify (a+b+c)3 — a (b + c − a) − b (a + c − b) − c (a+b−c).

37. Simplify (x-y)3 + (x + y)3+ 3 (x − y)3 (x + y) + 3 (x + y)3 (x − y). 38. Simplify (a+b2+c2)2 — (a+b+c)(a+b −c) (a+c−b)(b+c−a). 39. Simplify (a2+b2 +c3)2+(a+b+c)(a+b−c)(a + c −b)(b + c − a). 40. Prove that x3 + y3 + (x + y)3 = 2 (x2 + xy + y3)*

+ 8x2y2 (x + y)3 (x2 + xy + y2). 41. Prove that 4xy (x2 + y2) = (x2 + xy + y3)3 — (x2 — xy + y3)3. 42. Prove that 4xy (x2 − y3) = (x2 + xy — y3)3 — (x3 — xy — y2)2. 43. Multiply together (x-3x+2) and x2+ 6x + 1.

44. Multiply + a3 — ax (x3 + a3) by x3 + a3 — ax (x + a).

45. Multiply (a + b)3 by (a - b)3.

46. If s=a+b+c, prove that

s — 2b) (s — 2c) + 8 (s − 2c) (s — 2a) + s (s — 2a) (s -- 2b)

58. Division, as in Arithmetic, is the inverse of Multiplition. In Multiplication we determine the product arising from o given factors; in Division we have the product and one of e factors given, and our object is to determine the other factor. e factor to be determined is called the quotient.

59. Since the product of the numbers denoted by a and b denoted by ab, the quotient of ab divided by a is b; thus ÷a=b; and also ab÷b=a. Similarly, we have abc÷a = bc, ÷b= ac, abc÷c=ab; and also abc÷bc = a, abc÷ac = b,

÷ ab= c.

These results may also be written thus:

60. Suppose we require the quotient of 60abc divided by 3c. Since 60abc20ab × 3c we have 60abc÷ 3c = 20ab. Similarly, 60abc4a15bc; 60abc÷5ab12c; and so on. Thus we may

deduce the following rule for dividing one simple term by another: If the numerical coefficient and literal product of the divisor be found in the dividend, the other part of the dividend is the quotient.

61.

If the numerical coefficient and literal product of the divisor be not found in the dividend, we can only indicate the division by the notation we have appropriated for that purpose. Thus if 5a is to be divided by 2c, the quotient can only be indiIn some cases we may however

cated by 5a2c, or by

Ба 2c

simplify the expression for the quotient by a principle already used in Arithmetic. Thus if 15a3b is to be divided by 6bc, the quotient is denoted by

15a2b
6bc

Here the dividend = 36 × 5a3, and

the divisor = 36 × 2c; thus in the same way as. in Arithmetic we may remove the factor 3b, which occurs in both dividend and

divisor, and denote the quotient by

5a2 2c

62. One power of any number is divided by another power of the same number by subtracting the index of the latter power from the index of the former.

Thus aa2 = a × a × a ×a×a ÷ a × a = a × a × a = a3 = a5-2. Similarly any other case may be established.

Hence if m and n be any whole numbers, and m greater

63. Again, suppose we have such an expression as

the common factor a2. Thus we obtain

Similarly any

other case may be established. Hence if m and n be any whole

numbers, and m less than n, we have a"÷a" or

m α

α

n

=

1

m

to occur; this may be

α α

means X and X =
b b'

b

a3

as we

know from Arithmetic, and as will be shewn in Chapter VIII. Similarly any other case may be established.

65. When the dividend contains more than one term, and the ivisor contains only one term, we must divide each term of the ividend by the divisor, and then collect the partial quotients to obin the complete quotient.

abc + abd

ab

=b-c+d; for (b − c + d) ab = ab3 — abc + abd.

In the first example we see that corresponding to the term ab the dividend and to the divisor b there is the term a in the otient; and corresponding to the term cb in the dividend d to the divisor b there is the term c in the quotient.

We have already stated in Art. 49, that the following results - admitted for the present, subject to future explanation,

Thus in Division as in Multiplication, the sign of the quotient is deduced from the signs of the dividend and divisor by the rule, like signs produce +, and unlike signs produce -.

66. When the divisor as well as the dividend contains more than one term, we must perform the operation of algebraical division in the same way as the operation called Long Division in Arithmetic. The following rule may be given :

Arrange both dividend and divisor according to the powers of some common letter—either both according to ascending powers, or both according to descending powers. Find how often the first term of the divisor is contained in the first term of the dividend, and write down this result for the first term of the quotient; multiply the whole divisor by this term, and subtract the product from the dividend. Bring down as many terms of the dividend as the case may require, and repeat the operation till all the terms are brought down.

The reason for the rule is, that the whole dividend may be divided into as many parts as may be convenient, and the complete quotient is found by taking the sum of all the partial quotients. Thus, in the example, a2 - 2ab+b2 is really divided by the process into two parts, namely, a2 - ab and ab+b2, and each of these parts is divided by a-b; thus we obtain the complete quotient a-b.

67. It may happen, as in Arithmetic, that the division cannot be exactly performed. Thus, for example, if we divide a2 - 2ab+2b2 by a-b, we shall obtain as before a

- b in the