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, 12a1b3cd divided by 8a2bc2 gives 12a+b2cd 8 a2bc2 which may be reduced by dividing the numerator and denominator by the common factors, 4, a2, b, and c, giving 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. 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 = a11 = ao, a2 = a22 = ao, === a3 3 = ao, etc. α a2 a3 a3 α But since any quantity divided by itself is equal to 1, it follows that or, finally, if we designate the exponent by m, we have 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 6a2b2c1 by 2a2b2. 6 a2b2c1 =3a2-262-2c=3abc3c*. 2. Divide 8ab3c5 by -4a'b'c. Ans. 2a b°c* — — 2c*. 3. Divide 32m3n'x'y' by 4 m3n2xy. = 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, 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. 4. In 5ay-x-2 get rid of the negative exponents. 4a2b3x-2 5. In Ans. 3x2 get rid of the negative exponents. 3a-36-5 15a-8c-4d-5 6. In 45x-3y-5-2 get rid of the negative exponents. x3y5 3 a3c2d5 15a-4bc-1 9. In 5a-26-1 get rid of the negative exponents. Ans. 9a-1b-1, or 36' Ans. a2c Ans. 3 ab'c'. 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. |