CASE II. 101. When the given quantities can not be factored by inspection. 102. The greatest common divisor is found in this case by a process of decomposing the quantities by division. But in order to deduce a rule for the method, it will be necessary first to establish certain principles relating to exact division. 103. First, suppose A to be a quantity which is exactly divisible by another quantity, D, and let q represent the quotient. Then, А =9 D If we now multiply the dividend by m, we shall have, from (84 I), Am -=qm D in which qm is entire. Thus we have shown that if D divides A, it will also divide Am. Hence, 1. If a quantity will exactly divide one of two quantities, it will divide their product. Again, let A and B represent any two quantities, and S their sum. Now suppose both A and B are exactly divisible by D, and let A B F9, and Ea' We shall have A+B=S s 9+g'= = D S in which must be entire, because its equal,q+a', is entire. Hence, D 2. If a quantity will exactly divide each of two quantities, it will divide their sum. Finally, let d.represent the difference of A and B, and suppose A and B to be divisible by D, q and q' being the quotients, as before. We shall have AB=d And dividing every term by D, d 9-9'=0 in which is entire, because q-9' is entire. Hence, D 3. If a quantity will exactly divide each of two quantities, it will divide their difference. 104. We may now show, by the aid of these principles, wh relation the greatest common divisor of two quantities bears to tho parts of these quantities when decomposed by division. Suppose two polynomials to be arranged according to the powers of the same letter, and let A represent the greater and B the less. Then let us divide the greater by the less, the last divisor by the last remainder, and so on, till nothing remains. If we represent the several quotients by 9, 4, 9", etc.; and the remainders by R, R', R", etc., the successive operations will appear as follows : To investigate the mutual relations of A, B, R, and R', we observe that in division the product of the divisor and quotient, plus the remainder, if any, is always equal to the dividend. Hence, from the operations above, we have the three following conditions : R'q'' =R RY +R'=B B +R=A Now from the first equation it is evident that R' divideš R without remainder; it will therefore divide Rq', (103, 1). And since R' divides both Rq' and itself, it must divide their sum, Rq'+R', or B: (103, 2); consequently, it will divide Bq, (103, 1). Finally, since it divides both Bq and R, it must divide their sum, Bq+R, or A, (103, 2). Hence, I. The last divisor, R', is a common divisor of R, B, and A; or of all the dividends. Again, the dividend minus the product of the divisor and quotient, is always equal to the remainder. Therefore, from the first and second operations above, we have A-Bq=R Now any expression which will divide B, will divide Bq, (103, 1); hence, any expression which will divide both A and B, will also divide A-Bq, or R, (103, 3). Whence it follows that the greatest common divisor of A and B will divide R, and is therefore a common divisor of B and R. For like reasons, referring to the second equation, the greatest common divisor of B and R will also divide R', and is therefore a common divisor of R and R'. But the greatest common divisor of R and R' is R' itself. Consequently, R' is the greatest common divisor of R and B, and also of B and A. Hence II. The last divisor, R', is the greatest common divisor of the given quantities, and also of the dividend and divisor in each subsequent operation. 1. What is the greatest common divisor of 12x' —20°—7x43 and 3x*_22_1? FIRST OPERATION. 6x -3x4 -3 x-1 1st Rem. SECOND 'OPERATION. -1 3x-23-1 -1 Ans. X-1. The process here employed for finding the greatest common divisor of two polynomials, is subject to two modifications, which we will now investigate in their order. 1st. Suppressing monomial factors. It is evident that any monomial factor common to the given polynomials, may be suppressed in both, and set aside as one factor of their greatest common divisor. We may then apply the process of division to the resulting polynomials, and obtain the remaining factor or factors of the greatest common divisor required. Again, if either polynomial contains a factor which is not common to both, this factor can form no part of the greatest common divisor required, and may therefore be suppressed. And since the greatest common divisor of the given polynomials is the same as that of the dividend and divisor in each operation following the first, (II), it is evident that we may suppress the monomial factors in every remainder that occurs. And it should be observed, that if all the monomial factors of the given quantities have been previously suppressed, no monomial factor of any one of the remainders can belong to the greatest common divisor sought, or be common to any two successive remainders, (II). This modification of the process will be illustrated by the example which follows: 2. What is the greatest common divisor of 12x: +22.x* +6x and 6.6- 15x: _36x ? The first polynomial contains the monómial factor 2x, and the second contains the monomial factor 3x. We therefore suppress these factors, setting aside x, which is common, as one factor of the greatest common divisor sought. We then apply the process of division to the resulting polynomials, as follows: FIRST OPERATION. 26x? +39 Suppressing the factor 13 in this remainder, we have 22° +3 for the next divisor. SECOND OPERATION. aco4 Taking the last divisor, and the common factor, x, which was set aside at the beginning, we have (2x+3) Xx=2x+3x. Ans. 2d. Introducing monomial factors. It may happen at any stage of the process, that after suppressing every monomial factor of the divisor, its first term will not be exactly contained in the first term of the dividend. In such cases, the dividend may be multiplied by such a factor as will render its first divisible by the first term of the divisor. No factor thus introduced can be common to the dividend and divisor, since by hypothesis all the monomial factors of the divisor have previously been suppressed. Consequently, if the process of division be continued under this modification, the last divisor must be the greatest common divisor sought. This point will be illustrated by the following example : 3. What is the greatest common divisor of 2x*—12x+17xo+ 6.0–9 and 4x-18x* +19.c—3? We first multiply the greater polynomial by 2, to render its first term divisible by the first term of the other polynomial. FIRST OPERATION. 6x +15x*+15x-18 2x+ 5x + 5.3- 6 -82+29.0–15 In the above operation, we suppress the factor 3 in the first remainder, and multiply the result by 2, to render the first term divisible by the first term of the divisor. We thus obtain —4x° + 10.0"+103-12 for the second dividend. As the two partial quotients, x and -1, have no connection, they are separated by a comma. Multiplying the last divisor by 2 for a new dividend, we proceed as follows: SECOND OPERATION. 8x*—36x2 + 38x 6 -8x' +29.2--15 , +7 - 19x+ 57 Dividing this remainder by -19, we have 2–3 for the next divisor. |