Divide a3 + b3+c3-3abc by a+b+c.

Arrange the dividend according to descending powers

of a.

It will be seen that we arrange these terms according to descending powers of a; then when there are two terms, such as a2b and a2c, which involve the same power of a, we select a new letter, as b, and put the term which contains b before the term which does not; and again, of

the terms ab2 and abc, we put the former first as involving the higher power of b.

This example might also be worked, with the aid of brackets, thus:

a+b+c) a3

−3abc+b3 +c3 (a2 — a(b + c) + b2 — bc + c2

a3 + a2(b+c)

— a2(b+c)−3abc + b3 + c3

− a2(b+c) − a(b2 + 2bc+ c2)

a(b2- bc+c2) + b3 + c3

a(b2- bc+c2) + b3 + c3

Divide x3-(a+b+c)x2+(ab + ac+bc)x−abc by x-c.

x−c) x3 −(a+b+c)x2 + (ab+ ac+bc)x− abc (x2 — (a+b)x+ab x3- cx2

− (a+b)x2+(ab+ ac+bc)x− abc

− (a+b)x2 + (a+b)cx

Every example of Multiplication, in which the multiplier and the multiplicand are different expressions, will furnish two exercises in Division; because if the product be divided by either factor the quotient should be the other factor. Thus from the examples given in the section on Multiplication the student can derive exercises in Division, and test the accuracy of his work. And from any example of Division, in which the quotient and the divisor are different expressions, a second exercise may be obtained by making the quotient a divisor of the dividend, so that the new quotient ought to be the original divisor.

Divide

EXAMPLES. IX.

1. 15x3 by 3x2. 2. 24a by - 8a3. 3. 18x3y2 by 6x2y. 4. 24a4b5c6 by -3a2b3c4. 5. 20ab4x3y by 5b2x3y.

6. 4x3-8x2+16x by 4x. 7. 3a4-12a3 +15a2 by -3a2. 8. x3-3x2y2+4xy3 by xy.

9. -15a3b3 — 3a2b2+12ab by −3ab.

10. 60a3b3c2-48a2b4c2 +36a2b2c4-20abc by 4abc2.

11. x2-7x+12 by x−3.

12.

x2+x-72 by x+9.

13. 2x3-x2+3x-9 by 2x-3.

14. 6x3+14x2 - 4x + 24 by 2x+6. 15. 9x3-3x2+x-1 by 3x – 1.

16. 7x3-24x2 + 58x-21 by 7x-3.

17. 6-1 by x-1. 18. a3-2ab+b3 by a−b. -81y by x-3y.

19.

20. x1-2x3y+ 2x2y2 — xy3 by x—y.

21. x3-y5 by x-y.

22. a5 +32b5 by a +2b.

23. 2a+27ab3 - 81b4 by a+3b.

24. x5+ xy + x3y2 + x2y3 + xy1+y3 by x3 +y3.

25. x+2x1y+ 3x3y2 — x2y3 — 2xy1 — 3y3 by x3 — y3.

26. x-5x+11x2-12x+6 by x2-3x+3.

27. x2+x3-9x2-16x-4 by x2+4x+4.

28. x-13x2 +36 by x2+5x+6.

29. +64 by x2+4x+8.

30. x2+10x3 +35x2+50x +24 by x2 + 5x + 4.

31. x+x3-24x2 - 35x+57 by x2+2x-3. 32. 1-x-3x2-x5 by 1+2x+x2.

33. x6-2x3+1 by x2-2x+1.

34. a*+2a2b2+9b1 by a2 − 2ab+3b2.

36. x6+2x5 - 4x1 — 2x3 + 12x2 -2x-1 by x2 + 2x − 1.

37. x+2x6+3xa + 2x2 + 1 by x1- 2x3 + 3x2 - 2x + 1.
38. 12+x6-2 by x2+x2+1.

39. x3-(a+b+c) x2 + (ab+ ac+bc) x-abc

by x2- (a+b) x+ab.

40. a2x2+(2ac-b2) x2 + c2 by ax2 −bx+c.

41. x-x3y-xy3+y1 by x2 + xy + y2.

42. x3-3xy-y3-1 by x-y-1.

43. 49x2+21xy + 12yz - 16≈2 by 7x+3y-4z.

44. a2+2ab+b2 — c2 by a + b − c.

45. a3 +8b3+c3 - 6abc by a2 + 4b2 + c2 — ac − 2ab – 2bc. 46. a3+3a2b+3ab2 + b3 + c3 by a+b+c.

47. a2 (b+c)+b2 (a−c) + c2 (a−b) + abc by a+b+c. 48. x3-2ax2+(a2 + ab − b2)x− a2b + ab2 by x −

- a+b.

X. General Results in Multiplication.

79. There are some examples in Multiplication which occur so often in algebraical operations that they deserve especial notice.

The first example gives the value of (a + b) (a + b), that

is of (a+b); thus we have

(a+b)2=a2+2ab+b2.

Thus the square of the sum of two numbers is equal to the sum of the squares of the two numbers increased by twice their product.

Again, the second example gives

(a - b)2= a2-2ab+b2.

Thus the square of the difference of two numbers is equal to the sum of the squares of the two numbers diminished by twice their product.

The last example gives

(a+b) (a - b) a2-b2.

Thus the product of the sum and difference of two numbers is equal to the difference of their squares.

80. The results of the preceding Article furnish a simple example of one of the uses of Algebra; we may say that Algebra enables us to prove general theorems respecting numbers, and also to express those theorems briefly.

For example, the result

(a+b) (a - b)=a2-b2

is proved to be true, and is expressed thus by symbols more compactly than by words.

A general result thus expressed by symbols is often called a formula.

81. We may here indicate the meaning of the sign = which is made by combining the signs + and -, and which is called the double sign.

Since (a+b)2=a2+2ab+b2, and (a−b)2= a2-2ab+b2, we may express these results in one formula thus:

(a+b)2=a2+2ab+b3,

where indicates that we may take either the sign + or the sign, keeping throughout the upper sign or the lower sign. ab is read thus, "a plus or minus b."