- Suppose, however, that when we divide R into B, to ascertain if it will exactly divide it, we find that the quotient is Q, with a remainder, R'. Now, it has been shown, that whatever exactly divides two quantities, will divide their remainder after division; then, since the greatest common divisor of A and B, has been shown to divide B and R, it will also divide their remainder R', and can not be greater than R. And, if R' exactly divides R, it will also divide B, since B=RQ'+R'; and whatever exactly divides B and R, will also exactly divide A, since A=BQ+R; therefore, if R' exactly divides R, it will exactly divide both A and B, and will be their greatest common divisor. In the same manner, by continuing to divide the last divisor by the last remainder, it may always be shown, that the greatest common divisor of A and B will exactly divide every new remainder, and, of course, can not be greater than either of them. It may, also, always be shown, as above, in the case of R', that any remainder, which exactly divides the preceding divisor, will also exactly divide A and B. Then, since the greatest common divisor of A and B can not be greater than this remainder, and, as this remainder is a common divisor of A and B, it will be their greatest common divisor sought. To illustrate the same principle by numbers, let it be required to find the greatest common divisor of 14 and 20. 14 6)14(2 12 If we divide 20 by 14, and there is no remain- 14)20(1 der, 14 is, evidently, the greatest common divisor, since it can have no divisor greater than itself. Dividing 20 by 14, we find the quotient is 1, and the remainder 6, which is, necessarily, less than either of the quantities, 20 and 14; and by the theorem, Article 98, it is exactly divisible by their greatest common divisor; hence, the greatest common divisor must divide 20, 14, and 6, and cannot be greater than 6. Now, if 6 will exactly divide 14, it will also exactly divide 20, since 20=14+6, and will be the greatest common divisor sought. 6 But when we divide 6 into 14, to ascertain if it will exactly divide it, we find that the quotient is 2, with a remainder, 2; then, REVIEW.-95. When can a quadratic trinomial be separated into binomial factors? 96. What is the principal use of factoring? 97. What is a common divisor of two or more quantities? Give an example. 98. What is the greatest common divisor of two quantities? Give an example. 99. When are quantities commensurable? When are quantities incommensurable? 100. How do you find the greatest common divisor of two or more monomials? 101. Prove that any common divisor of two quantities will always exactly divide their remainder, after division. by the preceding theorem, the greatest common divisor of 14 and 6 will also divide 2, and therefore, can not be greater than 2. Now, if 2 will exactly divide 6, it will, also, exactly divide 14, since 14=6×2+2; and whatever will exactly divide 6 and 14, will also divide 20. But 2 exactly divides 6; hence it is the greatest common divisor of 14 and 20. ART. 103.-When the remainders decrease to unity, or when we arrive at a remainder which does not contain the letter of arrangement, we conclude that there is no common divisor to the quantities. ART. 104.—If one of the quantities contains a factor not found in the other, it may be canceled without affecting the common divisor (see example 3); and if both quantities contain a common factor, it may be set aside as a factor of the common divisor; and we may proceed to find the greatest common divisor of the other factors of the given quantities. This is self-evident. See Example 2. ART. 105.—We may multiply either quantity, by a factor not found in the other, without affecting the greatest common divisor. 2abx Thus, in the fraction the greatest common divisor of the 3abc' two terms, is evidently ab. Here, we may cancel the factors 2 and x in the numerator, or 3 and c in the denominator, without affecting the common divisor; for the common divisor of or 3abc' ab If we multiply the dividend by 4, a factor not found in the divisor, we have 8abx of which the common divisor is still ab. In the same manner we may multiply the divisor by any factor not found in the dividend, and the common divisor will still remain the same. If, however, we multiply the numerator by 3, which is a factor of the denominator, the result is of which the greatest com 6abx 3abc' mon divisor is 3ab, and not ab as before. Hence, we see, that the greatest common divisor will be changed, by multiplying one of the quantities by a factor of the other. REVIEW.-102. Show, that by dividing the last divisor by the last remainder, the greatest common divisor of two polynomials will exactly divide both the first and second remainders after division. ART. 106.-In the general demonstration, Art. 101, it has been shown, that the greatest common divisor of two quantities, also exactly divides each of the successive remainders; hence, the preceding principles apply to the successive remainders that arise, in the course of the operations necessary to find the greatest common divisor. The preceding principles will be illustrated by some examples. 1. Find the greatest common divisor of x3-y3 and xa—x2y2. Here the second quantity contains 2 as a factor, but it is not a factor of the first; we may, therefore, cancel it, and the second quantity becomes x2-y2. Divide the first by it. After dividing, we find that y2 is a factor of the remainder, but not of x2-y2, the dividend. Hence, by canceling it, the divisor becomes x-y; then, dividing by this, we find there is no remainder; thereforex-y is the greatest common divisor. 3-3 |x2—y2 x3-xy2 xy2-y3 or, (x-y)y2 (x x2-y2 x-y xy-y2 xy-y2 2. Find the greatest common divisor of x6+a3x3 and xa—a2x2. The factor 2 is common to both these quantities; it therefore forms part of the greatest common divisor, and may be taken out and reserved. Doing this, the quantities become x1+a3x and x2—a2. The first quantity still contains a common factor, x, which the latter does not; canceling this, it becomes a3+a3. Then, proceeding as in the first example, we find the greatest common divisor is x2(x+a). 2-3+a3 |x2—a2 (x a2x+a3 or, (x+a)a2 x2-a2 x+a x2+ax -ax-a2 —ax-a2 Xx a 3. Find the greatest common divisor of 5a3+10a*x+5a3x2 and a3x+2a2x2+2ax3+x1. Here 53 is a factor of the first quantity only, and x, of the second only. Suppressing these factors, and proceeding as in the previous examples, we find a+x is the greatest common divisor. a3+2a2x+2ax2+x3 \a2+2ax+x2 ax2+x3 (a ax+x2 ax-+x2 4. Find the greatest common divisor of 2a-a2x2-6x4 and 4a5+6a3x2-2a2x3-3x2. In solving this example, 4a5+6a3x2-2a2x3-3x2 there are two instances in which it is necessary to multiply the dividend, in order that the coefficient of the first term may be exactly divisible by the divisor. See Art. 105. 2a1-a2x2-6x1 (2a 4a5—2a3x2-12ax1 or, (8a-2a2x+12ax2-3x2)x2 8a3x2-2a2x2+12ax+-3x5 The greatest common divisor is found to be 2a2+3x2. 2a1-a2x2-6x4 4 8a3-2a2x+12ax2—3x3| 2a2+3x2 -93.x1 From the preceding demonstrations and examples, we derive the RULE, FOR FINDING THE GREATEST COMMON DIVISOR OF TWO POLYNOMIALS. 1st. Divide the greater polynomial by the less, and if there is no remainder, the less quantity will be the divisor sought. 2d. If there is a remainder, divide the first divisor by it, and continue to divide the last divisor by the last remainder, until a divisor is obtained, which leaves no remainder; this will be the greatest common divisor of the two given polynomials. REMARKS.-102. Explain the principles used, in finding the greatest common divisor, by finding it for the numbers 14 and 20. 103. When do we conclude that there is no common divisor to two quantities? 104. How is the common divisor of two quantities affected, by canceling a factor in one of them, not found in the other? When both quantities contain a common factor, how may it be treated? 105. How is the greatest common divisor of two quantities affected, by multiplying either of them by a factor not found in the other? What is the rule for finding the greatest common divisor of two polynomials? How do you find the greatest common divisor of three or more quantities? NOTES.-1. When the highest power of the leading letter is the same in both, it is immaterial which of the quantities is made the dividend. 2. If both quantities contain a common factor, let it be set aside, as forming a factor of the common divisor, and proceed to find the greatest common divisor of the remaining factors, as in Example 2. 3. If either quantity contains a factor not found in the other, it may be canceled, before commencing the operation, as in Example 3. See Art. 104. 4. Whenever it becomes necessary, the dividend may be multiplied by any quantity which will render the first term exactly divisible by the diviSee Art. 105. sor. 5. If, in any case, the remainder does not contain the leading letter, that is, if it is independent of that letter, there is no common divisor. 6. To find the greatest common divisor of three or more quantities, first find the greatest common divisor of two of them; then, of that divisor and one of the other quantities, and so on. The last divisor thus found, will be the greatest common divisor sought. 7. Since the greatest common divisor of two or more quantities contains all the factors common to these quantities, it may be found most easily by separating the quantities into factors, where this can be done, by means of the rules in the preceding article. Find the greatest common divisor of the following quantities. ART. 107.—A multiple of a quantity is that which contains it exactly. Thus, 6 is a multiple of 2, or of 3; and 24 is a multiple of 2, 3, 4, &c.; also, 8a2b3 is a multiple of 2a, of 2a2, of 2a2b, &c.; and 4(a-x)y is a multiple of (a-x), of 2y, of 4y", &c. ART. 108.-A quantity that contains two or more quantities exactly, is a common multiple of them. Thus, 12 is a common multiple of 2 and 3; and 6ax is a common multiple of 2, 3, a and x. |