VI. DIVISION. 85. Division, in Algebra, is the process of finding one of two factors, when their product and the other factor are given. Hence, Division is the converse of Multiplication. 14 ab 7a Thus, the division of 14 ab by 7a, which is expressed (Art. 15), signifies that we are to find a quantity which, when multiplied by 7a, will produce 14 ab. 86. The Dividend is the product of the factors. 87. Since the dividend is the product of the divisor and quotient, it follows, from Art. 77, that: 一。 If the divisor is +, and the quotient is +, the dividend is +. 一. In other words, if the dividend and divisor are both +, or both, the quotient is + ; and if the dividend and divisor are one +, and the other -, the quotient is Hence, in Division as in Multiplication, 一. Like signs produce +, and unlike signs produce -. 88. Required the quotient of 14 ab divided by 7a. By Art 85, we are to find a quantity which, when multiplied by 7a, will produce 14 ab. That quantity is evidently 2b; hence That is, the coefficient of the quotient is the coefficient of the dividend, divided by the coefficient of the divisor. 89. Required the quotient of a divided by a3. We are to find a quantity which, when multiplied by a3, will produce a. That quantity is evidently a2; hence That is, the exponent of a letter in the quotient is equal to its exponent in the dividend minus its exponent in the divisor. 90. We derive from Arts. 87, 88, and 89 the following rule for the division of monomials : To the quotient of the coefficients annex the literal quantities, giving to each letter an exponent equal to its exponent in the dividend minus its exponent in the divisor. Make the quotient + when the dividend and divisor have like signs, and they have unlike signs. when Note. A literal quantity having the same exponent in the dividend and divisor, as d+ in Ex. 2, is canceled by the operation of division, and DIVISION OF POLYNOMIALS BY MONOMIALS. 91. The operation being simply the converse of Art. 81, we have the following rule: Divide each term of the dividend by the divisor, and conneci the results with their proper signs. EXAMPLES. 1. Divide 9 ab - 6a2c + 12a2bc by - 3a2. 6. 5a2bc-5 ab2c + 5 abc2 by - 5 abc. 7. 4x2 - 8x -14x + 2x2 - 63 by 2. 8. -12 ab - 30a2b + 108a"b" by - 6 ambn. 9. 20x4 - 12x2 - 28x by 4x. 10. a2bc - ab2c2 + a2bc2 by - abc. 12. 15 xmy"z" - 35xm+2y2nz by 5xmy"z. 5 ab. DIVISION OF POLYNOMIALS BY POLYNOMIALS. 92. Required the quotient of 12 + 10x 11 x - 21 divided by 2x2 - 4 - 3x. Arranging both dividend and divisor according to the descending powers of x (Art. 37), we are to find a quantity which, when multiplied by the divisor, 2x2-3x-4, will produce 10 – 21 x2 11x+12. It is evident, from Art. 82, that the term containing the highest power of x in the product, is the product of the terms containing the highest powers of x in the factors. Hence 10x is the product of 2x2 and the term containing the highest power of x in the quotient. Therefore the term containing the highest power of x in the quotient is 10 divided by 2x2, or 5x. Multiplying the divisor by 5x, we have the product 10x – 15x2 - 20x; which, when subtracted from the dividend, leaves the remainder 6x2 + 9x + 12. This remainder is the product of the divisor by the rest of the quotient; hence, to obtain the next term of the quotient, we proceed as before, regarding - 6x2+ 9x +12 as a new dividend. Dividing the term containing the highest power of x, 6x2, by the term containing the highest power of x in the divisor, 2x2, we have the quotient. 3 as the second term of 6x2+9x+12; Multiplying the divisor by - 3, we have which, when subtracted from the second dividend, leaves no remainder. Hence 5x-3 is the required quotient. It is customary to arrange the work as follows: 10x - 21 x2 - 11 x + 12 | 2x2-3x-4, Divisor. 10 - 15 x2 - 20x 5x-3, 6x2+ 9x+12 6x2+ 9x+12 Quotient. Note. We might have solved the example by arranging the divi dend and divisor according to the ascending powers of x, in which case the quotient would have appeared in the form -3+5x. 93. From Art. 92, we derive the following rule for the division of polynomials : Arrange both dividend and divisor in the same order of powers of some common letter. Divide the first term of the dividend by the first term of the divisor, giving the first term of the quotient. Multiply the whole divisor by this term, and subtract the product from the dividend, arranging the remainder in the same order of powers as the dividend and divisor. Regard the remainder as a new dividend, and proceed as before; continuing until there is no remainder. Note. The work may be verified by multiplying the quotient by the divisor, which should of course give the dividend. |