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20. – 88a566c8 by 11 a3bc6.
21. 77 a*yozt by – 11a*y*z*.
22. 84 a*b?c?d by – 42a*bcod.
23. – 88a%%?cs by 84568c6.
24. 1622 by - 8%.
25. – 88a"b2 by 11 amb.
26. 77 amin by — 11 a2b3.
27. 84 a®Gm by 42a"bo.
28. – 88 apba by 8a5bm.
29. 96 abr by 48 amb4.
30. 168x@yo by 12 x^ym.
31. 256 aboca by 16 a"bmc.

Ans. – 8a2b2c.

Ans. 7.

Ans. -- 2. Ans. — 11 ab.

Ans. 2x. Ans. 8an-mb. Ans. — 7am-261–3.

Ans. 2a8-nym —%. Ans. — 11 ap-nge-m.

Ans. 2al-n78Ans. 14 24 –myb-m. Ans. 16 al-ng3 m.2 -P.

48. It follows from the preceding rules that the exact division of monomials will be impossible

(1) When the coefficient of the dividend is not exactly divisible by that of the divisor.

(2) When the exponent of the same letter is greater in the divisor than in the dividend.

(3) When the divisor contains one or more letters not found in the dividend.

In any of the above cases the quotient will be expressed by a fraction.

A fraction is said to be in its simplest form when the numerator and the denominator do not contain a common factor: for example, 12 a*b'cd divided by 8 abc

ve 12a*b'cd

which may be reduced by dividing the numerator and denominator by the common factors, 4, a’, b, and c, giving

12a*bạcd _ 3a2bd als 25a%bd* _ 5a.

8a2bc?

Hence, for the reduction of a monomial fraction to its simplest form, we have the following rule:

Suppress every factor, whether numerical or literal, that is common to both terms of the fraction. The result will be the reduced fraction sought.

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49. In dividing monomials it often happens that the exponents of the same letter in the dividend and divisor are equal, in which case that letter may not appear in the quotient. It might, however, be retained by giving to it the exponent 0.

If we have expressions of the form

and apply the rule for the exponents, we shall have

=a'-=a", a = q*+=a", " = q**=co, etc. But since any quantity divided by itself is equal to 1, it follows that

=a° = 1, ' = 20-=ao=1, etc.; or, finally, if we designate the exponent by m, we have

a”=am-* = a° = 1; that is,

The O power of any quantity is equal to 1 : therefore .

Any quantity may be retained in a term, or introduced into a term, by giving it the exponent 0.

Exercises.

1. Divide 6 a2b2c4 by 2 ab?.

6a*b*c* = 3 a2-702–2* = 3 a ob°c* = 36.

2 ^?ba 2. Divide 8a+b3c6 by – 4a4b°c. Ans. — 2 aob°c=-2c*. 3. Divide – 32 mån?x?y? by 4 monxy.

Ans. - 8 monoxy = -8xy.

4. Divide – 96 a65cm by — 24 aʻ65.
5. Introduce a as a factor into 665c.
6. Introduce ab as factors into 9c%d".
7. Introduce abc as factors into 8d4fm

Ans. 4 a°°c" = 4c".

Ans. 6 a°65c*. Ans. 9 aob°cód". Ans. 8aob°cod4fm

50. When the exponent of any letter is greater in the . divisor than it is in the dividend, the exponent of that letter in the quotient may be written with a negative sign. Thus,

"=^-5 = a-3, by the rule :

hence a-s=

Since aO= , we have 6 xa-s=; that is, a in the numerator with a negative exponent is equal to a in the denominator with an equal positive exponent. Hence

Any quantity having a negative exponent is equal to the reciprocal of the same quantity with an equal positive exponent.

Any factor may be transferred from the denominator to the numerator of a fraction, or the reverse, by changing the sign of its exponent.

Exercises. 1. Divide 32 abc by 16 a56?.

32 abco -35-1

Ans. 16 2572

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3. Reduce

17xy? z 51x*y;

4. In 5 ay-z-get rid of the negative exponents.

Anse og -5 get rid of the negative exponents.

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Ans. 4a 78

Ans. 33

322 * In 15 a-8-4d-5 In 45x-8y-62-3 - get rid of the negative exponents.

*3 aocode 7. Reduce -8a-835c

Ans - 40-5810c . 14 aʼ6-6c

Ans.

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705 .

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51: To Divide a Polynomial by a Monomial.

Divide each term of the dividend separately by the divisor. The algebraic sum of the quotients will be the quotient sought.

Exercises. Divide 1. 3a2b? – a by a, 2. 50%b? 25 a*b? by 5a8%?. 3. 35 ab? — 25 ab by – 5 ab. 4. 10 ab — 15 ac by 5a. 5. 6 ab 8ax +4 aʻy by 2a.

D. N. E. A. --5.

Ans. 3 ab? — 1.

Ans. 1-5 a. Ans. 7 ab +5.

Ans. 26 3c. Ans. 36-4x+ 2 ay.

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