Note. If, in Ex. 2, the expressions had been arranged in ascending powers of x, it would have been found unnecessary to introduce
a numerical factor in the course of the work.
119. The use of the Factor Theorem (Art. 105) often lessens, in a very marked degree, the work of finding the highest common factor. Thus in Ex. 2 of the preceding article it is easily seen that both expressions become equal to when 1 is substituted for x, hence x - 1 is a factor. Dividing the first of the given expressions by x - 1, we obtain a quotient 2 x^2 +3x + 2. It is evident that this will not divide the second expression, hence x - 1 is the H. C. F.
120. When the Method of Division by Detached Co-efficients (Art. 63) is employed in finding the H. C. F., the following is a convenient arrangement. Ex. Find the H. C. F. of x4 + 3 x3 + 12x - 16 and x^3 -13x+ 12.
We write the literal factors of the dividend until we reach a term of the same degree as the first term of the divisor.
x^3 x^4 + 3x^3+ 0+12-16 + 13 + 13 - 12 - 12 + 39 - 36 x +3; 13 + 39 - 52
The addition of the terms in the third column gives 13 x^2, which is of lower degree than the first term of the divisor, hence we can proceed no further with the division and have for a remainder 13 x^2 + 39 x - 52. Removing from this remainder the factor 13, as it is not a factor of the given expressions, we have for a second divisor x'-^ + 3 x - 4. The first divisor, as written before the signs were changed, forms the second dividend :
x2 x3 + 0x^2 - 13 + 12 -3 -3+4 + 4 +9-12 x - 3 ; 0 0
since there is no remainder, the last divisor, as written before the signs, were changed is the H. C. F. Thus x^2 + 3 x - 4 is the H. C. F..
