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CHAPTER VI.

DIVISION.

46. THE object of division is to find out the quantity, called the quotient, by which the divisor must be multiplied so as to produce the dividend.

Division is thus the inverse of multiplication.
The above statement may be briefly written

quotient x divisor=dividend,

dividend - divisor=quotient. It is sometimes better to express this last result as a fracthus

dividend
divisor

quotient.

or

tion;

DIVISION OF SIMPLE EXPRESSIONS.

Example 1. Since the product of 4 and x is 4x, it follows that when 4x is divided by a the quotient is 4, or otherwise,

4x = x=4.

27a5 27ααααα Example 2. 27a5 : 9a3 =

= 3aa, by removing from

9a3 9aaa the divisor and dividend the factors common to both, just as in Arithmetic. Therefore

27a5 = 9a’= 3a?.

35aaabbccc Example 3. 35a3b2c3 = 7ab2c2=

7abbcc =5a2c.

47. RULE. To divide one simple expression by another, divide the coefficient of the dividend by that of the divisor, and subtract the index of any letter in the divisor from the index of that letter in the dividend.

a

ax-5

a

- a=

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Example 4. 84a x3 = 12a4x=7a5-423–1

=7axa. Example 5. 77a-x3y4 =- Tax-y=11axy3.

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. Thus, by the rule a3 = a=a3–3=a';

a3 but also

a3 = a:= =1,

a3

.. a?=1. This result will appear somewhat strange to the beginner, but its full significance will be explained in Chapter XXXI.

48. It is easy to prove that the rule of signs holds for division.

ab axb Thus ab ; a=

=b. a

- ab
- ab ; a=

-6.
a
ab

-ax-b
ab:

- a
- axb

=b.

-- a Hence in division as well as multiplication

like signs produce +,

unlike signs produce Examples. (1) 6ab: 2a=36.

(2) – 15xy =- 3x=- 5y.
(3) — 21a2b3 = - 7a2b2 =36.

(4) 45a662,4 : - 9a3bx2= - 5a3bx?.
DIVISION OF A COMPOUND EXPRESSION BY A SIMPLE

EXPRESSION. 49. RULE. To divide a compound expression by a single factor, divide each term separately by that factor.

This follows at once from Art. 34.
Examples. (1) (9x – 12y + 3x) = -3=- 3x + 4y 2.

(2) (36a3b2 - 24a15 20a4b) --4a+b=9ab 664 - 5a2b.
(3) (2xc? 5xy + 2x2y3) = - fx=- 4x + 104 3xy3.

- a

- ab

- ab ;

- a=

-a

EXAMPLES VI. a.

- Ax.

Divide 1. 3.2.3 by x2

2. 27x4 by - 9x3. 3. - 3526 by 7.x3.

4. abx2 by 5. XPy3 by xay.

6. a423 by - a223. 7. 4a2b2c3 by ab%ca.

8. 12a666c6 by – 3a4b2c. 9. - a5c9 by - ac.

10. 15xby74 by 5x2y2z2. 11. - 16.0"ya by – 4.xy

12. - 48a9 by-8a3. 13. 35ali by 7a".

14. 63a7b8c3 by 9a5b5c. 15. 7a2bc by - 7a-bc.

16. 28a183 by - 4a3b. 17. 166oy.x2 by - 2xy.

18. – 504323 by – 5.xoy. 19. 22 – 2xy by X.

20. 2.3 – 3x2 + x by x. 21. 26 – 7x5 +4.x* by x2. 22. 10x7 – 826 + 3x4 by 23. 23. 15.20 – 25x* by – 5x3. 24. 27.26 – 36.205 by 9x”. 25. – 24x6 – 32x4 by - 8x3. 26. 34.cy - 51.0y3 by 17.xy. 27. a? - ab ac by

28. a3 aub aj2 by a . 29. 3.23 – 9.x2y 12xy2 by – 3x. 30. 4x+y+ -- 8.x*?y2 +6.cy: by - 2.cy. 31. - 3a + ab-Gac by - a. 32.cy-3.c°yt by - coy. 33. - 22 + xy + 12x by- &c. 34. - 205x3 + Zafc4 by ja’x.

- a.

DIVISION OF COMPOUND EXPRESSIONS.
50. To divide one compound expression by another.

RULE. 1. Arrange divisor and dividend in ascending or descending powers of some common letter.

2. Divide the term on the left of the dividend by the term on the left of the divisor, and put the result in the quotient.

3. Multiply the WHOLE divisor by this quotient, and put the product under the dividend. 4. Subtract and bring down from the dividend as many terms

be

necessary. Repeat these operations till all the terms from the dividend are brought down.

as may

Example 1. Divide za +112 +30 by x+6.
Arrange the work thus:

2C+6)x2 + 11x + 30 ( divide æ, the first term of the dividend, by x, the first term of the divisor; the quotient is x. Multiply the whole divisor by x, and put the product 2% + 6x under the dividend. We then have

2C+6)x2 +11x + 30(xc

2? + 6x by subtraction

5x + 30. On repeating the process above explained we find that the next term in the quotient is +5. The entire operation is more compactly written as follows:

2C+ 6 )ca +11x + 30 (2C+5
202 + 6.

5x + 30
5x + 30

The reason for the rule is this: the 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 22+11x+30 is divided by the above process into two parts, namely x2 +62, and 5x+30, and each of these is divided byx+6; thus we obtain the complete quotient x+5. Example 2. Divide 2422 – 65xy +21y2 by 8x – 3y.

8x – 3y ) 24x8 – 65xy + 2172 ( 32 – 7y

24x2 – 9xy

- 56xy +2172
- 56xy + 2142

EXAMPLES VI. b. Divide 1. x2 + 3x + 2 by x+1.

2. 22 — 7x+12 by x-3. 3. a? – lla+30 by a - 5. 4. a?– 49a +600 by a – 25. 5. 3.x2 +10x +3 by X+3. 6. 202 + 11x +5 by 2x+1. 7, 5x2+11x + 2 by x+2, 8. 2x2 + 17x+21 by 2x+3. 9. 5x2 + 16x +3 by X+3. 10. 3.x2 +34x+11 by 3x+1. 11. 4.x2 +23x + 15 by 4x+3. 12. 6x2 — 73 — 3 by 2.0 – 3.

13. 3x2 + x – 14 by x - - 2.
14. 3.x2 – 3 – 14 by x+2.
15. 622 – 31x+35 by 2x 7.
16. 4x2 + x – 14 by x + 2.
17. 12a2 Tax 12x2 by 3a – 4.x.
18. 15a2 +17ax – 4.x2 by 3a +4x.
19. 12a2 – llac – 36c2 by 4a – 9c.
20. 9a+6ac – 35c2 by 3a+7c.
21. 60x2 – 4xy 45y2 by 10.x – 9y.
22. – 4xy 15y2 +96x2 by 12x 5y.
23. 723 +96x4 – 28x by 7x – 2.
24. 100.x2 – 3x – 13x2 by 3+25x.
25. 27203 +9x2 – 3x – 10 by 3.0 — 2.
26. 1643 – 46a2 +39a – 9 by 8a-3.
27. 15+3a – 7a2 – 4a3 by 5 – 4a.
28. 16 – 96x + 216x2 – 216x3 +81x4 by 2 – 3x.

51. The process of Art. 50 is applicable to cases in which the divisor consists of more than two terms. Example. Divide 6x5 -- x4 +4.23 – 5x2 – X – 15 by 2x2 – X+3. 2x2 – x+3)6x5 – 24 + 4x2 – 5x2 – – – 15(3.x3 + x2 – 2x – 5

6x5 – 3x4 + 9x3

2x4 – 5x3 – 5.x?
2x4 – x3 + 3x2

- 4x3 - 8.02
- 4.x3 + 2x2 - 6x

- 10.x2 + 5x - 15
- 10x” + 5x – 15

52. Sometimes it will be found convenient to arrange the expressions in ascending powers of some common letter. Example. Divide 223 +10 – 16a – 39a2 +15a4 by 2 - 4a – 5a2. 2 - 4a – 5a2,10 – 16a – 39a2 +223 +15a4 (5+24 – 3a2

10 - 20a - 25a

40 – 14a2+ 2a3
4a - 8a2 - 10a3

6a2 + 12a3 + 15a4
6a2 + 12a3 +1544

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