121. Let the two expressions in Ex. 2, Art. 118, be written in the form
2x^3 + x^2 - x - 2 =(x - 1)(2x^2 + 3x + 2), 3x^3 - 2x^2 +x-2 =(x - 1)(3x^2 + x + 2).
Then their highest common factor is x — 1, and therefore 2 x^2 +3x+2 and 3 x^2 + x +2 have no algebraic common divisor. If, however, we put x=6, then
2x^3 + x^2+x-2 = 460, and 3 x^3 - 2 x^2 + x - 2 = 580 ;
and the greatest common measure of 460 and 580 is 20; whereas 5 is the numerical value x — 1, the algebraic highest common factor. Thus the numerical values of the algebraic highest common factor and of the arithmetical greatest common measure do not in this case agree.
The reason may be explained as follows : when x = 6, the expressions 2 x^2 + 3 x + 2 and 3 x^2+ x + 2 become equal to 92 and 116 respectively, and have a common arithmetical factor 4; whereas the expressions have no algebraic common factor.
It will thus often happen that the highest common factor of two expressions and their numerical greatest common measure, when the letters have particular values, are not the same ; for this reason the term greatest common measure is inappropriate when applied to algebraic quantities.
EXAMPLES XI. c.
Find the highest common factor of the following expressions :
1. x^3 + 2x^2-13x+10, x^3 + x^2- 10x + 8. 2. x^3 - 5x^2 - 99x + 40, x^3 - 6x^2 - 86 x + 35. 3. x^3 + 2 x^2 - 8 x - 16, x^3 + 8 x^2 - 8 x - 24. 4. x^3 - x^2 - 5x - 3, x^3 - 4 x^2 - 11 x - 6. 5. x^3 + 3x^2 - 8x - 24, x^3 + 3 x^2 - 3x - 9. 6. a^3 - 5 a^2x + 7 ax^2 , a^3 - 3 ax^2 + 2 x^3. 7. 2x^3-5x^2+ 11x + 7, 4x^3- 11x^2 + 25x+ 7. 8. 2 x^3 + 4 x^2 - 7 x - 14, 6 x^3 - 10 x^2 - 21 x + 35.
