Divide 25. x+y3 −2x3y3 by (x−y)3. 26., x+y+2x3y3 by (x+y)2. 27. (a3-3a2b+5ab2 — 3b3) (a−2b) by a2¬3ab+262. 28. (x3-9x2y+23xy2 – 15y3) (x−7y) by x2-8xy + 7y2. 29. a+ab+b8 by (a2-ab+b2) (a2+ab+b2). 30 a8-b8+ a2b2 (a-b4) by (a2-ab+b2) (a2+ab+b2). 31. 4ab2+2(3a-2b)-ab (5a3-1162) by (3a-b)(a+b). 32. (x2-3x+2) (x-3) by x2- 5x + 6. 33. (x3-3x+2)(x+4) by x2+x-2. 34. (a2+ax+x2) (α3+x3) by a1+a2x2+x2. 35. (a+a2b2+b) (a+b) by a2+ab+b2. 36. b(x+a3)+ax (x2 — a3) + a3 (x+a) by (a+b)(x+a). XII. Greatest Common Measure. 97. In Arithmetic a whole number which divides another whole number exactly is said to be a measure of it, or to measure it; a whole number which divides two or more whole numbers exactly is said to be a common measure of them. In Algebra an expression which divides another expression exactly is said to be a measure of it, or to measure it; an expression which divides two or more expressions exactly is said to be a common measure of them. 98. In Arithmetic the greatest common measure of two or more whole numbers is the greatest whole number which will measure them all. The term greatest common measure is also used in Algebra, but here it is not very appropriate, because the terms greater and less are seldom applicable to those algebraical expressions in which definite numerical values have not been assigned to the various letters which occur. It would be better to speak of the highest common measure, or of the highest common divisor; but in conformity with established usage we shall retain the term greatest common measure. The letters G. C. M. will often be used for shortness instead of this term. We have now to explain in what sense the term is used in Algebra. 99. It is usual to say, that by the greatest common measure of two or more simple expressions is meant the greatest expression which will measure them all; but this definition will not be fully understood until we have given and exemplified the rule for finding the greatest common measure of simple expressions. The following is the Rule for finding the G.C.M. of simple expressions. Find by Arithmetic the G.C.M. of the numerical coefficients; after this number put every letter which is common to all the expressions, and give to each letter respectively the least index which it has in the expressions. 100. For example; required the G. C. M. of 16ab2c and 20a3b3d. Here the numerical coefficients are 16 and 20, and their G. C. M. is 4. The letters common to both the expressions are a and b; the least index of a is 3, and the least index of b is 2. Thus we obtain 4a3b2 as the required G. C.M. Again; required the G. c. M. of 8a2b3c2xyz3, 12abcx2y3, and 16a3c3x2y. Here the numerical coefficients are 8, 12, and 16; and their G. C. M. is 4. The letters common to all the expressions are a, c, x, and y; and their least indices are respectively 2, 1, 2, and 1. Thus we obtain 4a2ca2y as the required G.C.M. 101. The following statement gives the best practical notion of what is meant by the term greatest common measure, in Algebra, as it shews the sense of the word greatest here. When two or more expressions are divided by their greatest common measure, the quotients have no common measure. Take the first example of Art. 100, and divide the expressions by their G.c. M.; the quotients are 4ac and 5bd, and these quotients have no common measure. Again, take the second example of Art. 100, and divide the expressions by their G.C.M.; the quotients are 2b3cx3×3, 3a2bу2, and 4ac2y3, and these quotients have no و. common measure. 102. The notion which is supplied by the preceding Article, with the aid of the Chapter on Factors, will enable the student to determine in many cases the G.C.M. of compound expressions. For example; required the G. C. M. of 4a2(a+b)2 and 6ab(a2-b2). Here 2a is the G. C.M. of the factors 4a2 and 6ab; and a+b is a factor of (a+b)2 and of a2-b2, and is the only common factor. The product 2a(a+b) is then the G. C. M. of the given expressions. But this method cannot be applied to complex examples, because the general theory of the resolution of expressions into factors is beyond the present stage of e student's knowledge; it is therefore necessary to adopt another method, and we shall now give the usual definition and rule. 103. The following may be given as the definition of the greatest common measure of compound expressions. Let two or more compound expressions contain powers of some common letter; then the factor of highest dimensions in that letter which divides all the expressions is called their greatest common measure. 104. The following is the Rule for finding the greatest common measure of two compound expressions. Let A and B denote the two expressions; let them be arranged according to descending powers of some common letter, and suppose the index of the highest power of that letter in A not less than the index of the highest power of that letter in B. Divide A by B; then make the remainder a divisor and B the dividend. Again make the new remainder a divisor and the preceding divisor the dividend. Proceed in this way until there is no remainder; then the last divisor is the greatest common measure required. 105. For example; required the G. C. M. of x2-4x+3 and 4x3-9x2-15x+18. x2 - 4x+3)4x3- 9x2-15x+ 18 (4x+7 4x3-16x2+12x 7x2-27x+18 7x2-28x+21 X- 3 x-3)x2-4x+3 (x−1 x2-3x -x+3 -x+3 Thus -3 is the G. C. M. required. 106. The rule which is given in Art. 104 depends on the following two principles. (1) If P measure A, it will measure mA. For let a denote the quotient when A is divided by P; then AaP; therefore mA=maP; therefore P measures mA. (2) If P measure A and B, it will measure mA±nB. For, since P measures A and B, we may suppose A=aP, and B=bP; therefore mA±nB=(ma±nb)P; therefore P measures mA±nB. 107. We can now demonstrate the rule which is given in Art. 104. Let A and B denote the two expressions. Divide A by B; let p denote the quotient, and C the remainder. Divide B by C; let a denote the quotient, and D the remainder. Divide C by D, and suppose that there is no remainder, and let r denote the quotient. Thus we have the following results. A=pB+C, B) A (p pB C=rD. qC D) C (r rD We shall first shew that D is a common measure of A and B. Because C=rD, therefore D measures C; therefore, by Art. 106, D measures qC, and also qC+D; that is, D measures B. Again, since D measures B and C, it measures pB+C; that is, D measures A. Thus D measures A and B. We have thus shewn that D is a common measure of A and B; we shall now shew that it is their greatest common measure. By Art. 106 every common measure of A and B measures A-pB, that is C; thus every common measure of A and B is a common measure of B and C. Similarly, every common measure of B and C is a common measure |