Note. The term greatest common measure is sometimes used
instead of highest common factor ; but, strictly speaking, the term
greatest common measure ought to be confined to arithmetical quantities ; for the highest common factor is not necessarily the greatest
common measure in all cases, as will appear later. (Art. 121.)
116. We begin by working out examples illustrative of the algebraic process of finding the highest common factor, postponing for the present the complete proof of the rules we use. But we may conveniently enunciate two principles, which the student should bear in mind in reading the examples which follow.
I. If an expression contains a certain factor, any multiple of the expression is divisible by that factor.
II. If two expressions have a common factor, it will divide their sum and their difference ; and also the sum and the difference of any multiples of them.
Ex. Find the highest common factor of 4 x3 - 3 x2 - 24 x - 9 and 8 x3 - 2 x2 - 53 x - 39.
x 4 x3 - 3 x2 - 24 X -9 4x3-5x2-21x 2x 2x2 - 3x - 9 2x2 - 6x 3 3x-9 3x-9
8x3- 2x2 -53 x -39 8x3 - 6x2 -48 X -18 4x2 - 5 X — 21 4x2 - 6x- 18 x - 3
Therefore the H. C. F. is x - 3.
Explanation. First arrange the given expressions according to descending or ascending powers of x. The expressions so arranged having their first terms of the same order, we take for divisor that whose highest power has the smaller coefficient. Arrange the work in parallel columns as above. When the first remainder 4x2— 5 x— 21 is made the divisor we put the quotient x to the left of the dividend. Again, when the second remainder 2 x2 — 3 x — 9 is in turn made the divisor, the quotient 2 is placed to the right ; and so on. As in Arithmetic, the last divisor x — 3 is the highest common factor required.
117. This method is only useful to determine the compound factor of the highest common factor. Simple factors of the given expressions must be first removed from them,
