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How do you find the greatest common factor of a polynomial?

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To find the greatest common factor of a polynomial,  find a number that all the coefficients of the polynomial are divisible by. I.e. the GCF of a 9x^2 + 6x + 3  is 3 because all the coefficients (9, 6, 3) are divisible by 3 and the expression simplifies to 3(3x^2 + 2x + 1). 
While the gcf can be smaller than the smallest coefficient 
for example 20x^2 + 15x + 10 (gcf is 5, smallest coefficient is 10) 5<10
it CANNOT be larger than the smallest coefficient
since the smallest coefficient has to be larger than or equal to the gcf in order to be divisible by it.