CHAPTER VII. HIGHEST COMMON FACTORS. LOWEST COMMON 94. A Common Factor of two or more integral algebraical expressions is an integral expression which will exactly divide each of them. The Highest Common Factor of two or more integral expressions is the integral expression of highest dimensions which will exactly divide each of them. It is usual to write H.C.F. instead of Highest Common Factor. 95. The highest common factor of monomial expressions. The highest common factor of two or more monomial expressions can be found by inspection. Thus, to find the highest common factor of a3b2c and a1b1c3. The highest power of a which will divide both expressions is a3; the highest power of b is b2; and the highest power of c is c; and no other letters are common. Hence the H.C.F. is a3b2c. Again, to find the highest common factor of a3b4c4, a2b3 and a3bc2. The highest power of a which will divide all three expressions is a2; the highest power of b which will divide them all is b; and c will not divide all the expressions. Hence the H.C.F. is a2b. From the above examples it will be seen that the H.C.F. of two or more monomial expressions is the product of each letter which is common to all the expressions taken to the lowest power in which it occurs. 96. Highest common factor of multinomial expressions whose factors are known. Where the factors of two or more multinomial expressions are known, their H.C.F. can be at once written down, as in the preceding case. The H.C.F. will be the product of each factor which is common to all the expressions taken to the lowest power in which it occurs. Thus, to find the H.C.F. of (x − 2)3 (x − 1)2 (x − 3) and (x-2)2(x-1) (x-3)3. It is clear that both expressions are divisible by (x-2)2, but by no higher power of x- 2. Also both expressions are divisible by x − 1, but by no higher power of x-1; and both expressions are divisible by x-3, but by no higher power of x-3. Hence the н.C.F. is (x − 2)2 (x − 1) (x − 3). Again, the H.C.F. of a2b3 (a - b)2 (a+b)3 and a3b2 (a − b) (a + b)2 is a2b2 (a - b) (a+b)3. In the following examples the factors can be seen by inspection, and hence the H.C.F. can be written down. Ex. 1. Find the H.C.F. of a1b2 – a2b1 and aab3 + a3ba. Ans. a2b2 (a+b). Ex. 2. Find the H. C. F. of a6b2 - 4a4b4 and a6b2 - 16a2b6. Ans. a2b2 (a2-4b2). Ex. 3. Find the н.C.F. of a3+3a2b+2ab2 and a1+6a3b+8a2b2. Ans. a (a+2b). 97. Although we cannot, in general, find the factors of a multinomial expression of higher degree than the second [Art. 84], there is no difficulty in finding the highest common factor of any two multinomial expressions. The process is analogous to that used in arithmetic to find the greatest common measure of two numbers. If the expressions have monomial factors, they can be seen by inspection; and the highest common factors of these monomial factors can also be seen by inspection: we have therefore only to find the multinomial expression of highest dimensions which is common to the two given expressions. Let A and B stand for the two expressions, which are supposed to be arranged according to descending powers of some common letter, and let A be of not higher degree than B in that letter. Divide B by A, and let the quotient be Q and the remainder R; then Now an expression is exactly divisible by any other if each of its terms is so divisible; and therefore B is divisible by every common factor of A and R, and R is divisible by every common factor of A and B. Hence the common factors of A and B are exactly the same as the common factors of A and R; and therefore the H.C.F. of A and R is the H.C.F. required. Now divide A by R, and let the remainder be S; then the H. C. F. of R and S will similarly be the same as the H.C.F. of A and R, and will therefore be the H.C.F. required. And, if this process be continued to any extent, the H.C.F. of any divisor and the corresponding dividend will always be the H.C.F. required. If at any stage there is no remainder, the divisor must be a factor of the corresponding dividend, and that divisor is clearly the H.C.F. of itself and the corresponding dividend. It must therefore be the H. C. F. required. It should be remarked that by the nature of division the remainders are successively of lower and lower dimensions; and hence, unless the division leaves no remainder at some stage, we must at last come to a remainder which does not contain the common letter, in which case the given expressions have no H.C.F. Since the process we are considering is only to be used to find the highest common multinomial factor, it is clear that any of the expressions which occur may be divided or multiplied by any monomial expression without destroying the validity of the process; for the multinomial factors will not be affected by such multiplication or division. Thus the H.C.F. of two expressions can be found by the following Rule. Arrange the two expressions according to descending powers of some common letter, and divide the expression which is of the highest degree in the common letter by the other (if both expressions are of the same degree it is immaterial which is used as the divisor). Take the remainder, if any, after the first division for a new divisor, and the former divisor as dividend; and continue the process until there is no remainder. The last divisor will be the H. C.F. required. The process is not used for finding common monomial factors, these can be seen by inspection; and any divisor, dividend, or remainder which occurs may be multiplied or divided by any monomial expression. Ex. 1. Find the H.C.F. of x3+x2 - 2 and x3 + 2x2 - 3. The work would be shortened by noticing that the factors of x2-1, the first remainder, are x-1 and x+1. And by means of Art. 88 it is at once seen that x-1 is, and that x+1 is not, a factor of x3 + x2 - 2. Ex. 2. Find the H. C. F. of x3+4x2y - 8xy2+24y3 and x5 − x1y +8x2y3 — 8xy1. The second expression is divisible by x, which is clearly not a common factor: we have therefore to find the H.C.F. of the first expression and x1 − x3y + 8xу3 – 8y4. x3 + 4x2y − 8xy2+24y3) x − x3y +8xy3 — 8y1 x-5y - 5x3y +8x2y2 - 16xy3 - 8y1 28x2у2-56xу3 + 112y1 The remainder = 28y2 (x2 - 2xy +4y2): the factor 28y2 is rejected and x2 -2xy +4y2 is used as the new divisor. x3 x2 - 2xy +4y2) x3 + 4x2y − 8xy2+24y3 ( x +бy x3- 2x2y+4xy2 6x2y-12xy2+24y3 Hence x2-2xy +4y2 is the H.C.F. required. Ex. 3. Find the H.C.F. of 2x4+9x3 +14x+3 and 3x4+15x3+5x2 +10x+2. To avoid the inconvenience of fractions, the second expression is multiplied by 2: this cannot introduce any additional common factors. The process is generally written down in the following form: 2x1+9x3+14x+3) 3x1+15x3+5x2+10x+2 2 Thus x2+5x+1 is the H. C. F. required. Detached coefficients should generally be used [Art. 63]. 98. The labour of finding the H.C.F. of two expressions is frequently lessened by a modification of the process, the principle of which depends on the following Theorem :-The common factor of highest degree in a particular letter, a suppose, of any two expressions A and B is the same as the H.C.F. of pÅ +qB and rA +sB, |