Question

How do we find the greatest common factor of a polynomial? Demonstrate the process with an example, showing your work. When finding the greatest common factor of a polynomial, can it ever be larger than the smallest coefficient? Can it ever be smaller than the smallest coefficient?

 

Answer 1

umm here is an example:

5x^3 - 20x^ 2 - 25x

first you find something that can go into all of them, i chose this one because you can clearly see 5 is the higest thing that can go into all of them, not just 5 but what can you take out of all of them equally? 5x

5x ( x^2 - 4x - 5) now you factor it normally

5x (x - 5) ( x + 1) = 0

x = 0, 5, -1

you have to be able to take it out of them all equally, so to answer your question, it cannot be larger than the smallest coefficiant because even if it goes into the others equally, it still wont go into the smallest on equally if its larger.  Yes it can be smaller than the smallest coeficcient if like:

4x^3 + 6x^2 + 12x  here the greatest common factor is 2, which is smaller than the smallest coeficcient, as long as it goes into everything equally.