greatest common factor of terms

The greatest common factor of the terms of a polynomial is the biggest factor — numbers and variables together — that divides every single term evenly. It is the 'most you can take out' in one move, and removing it is the disciplined first step of any factoring problem, the analogue of finding the GCF of two integers before reducing a fraction.

To build it, handle numbers and variables separately. For the coefficients, take their greatest common divisor; for each variable that appears in all terms, take the lowest exponent it has anywhere. The product of those pieces is the GCF of the terms. For 8x^3y + 12x^2y^2, the number part is GCF(8, 12) = 4, and the shared variable part is x^2y, so the GCF is 4x^2y.

Pulling the GCF out and writing it in front of a parenthesis is just the distributive property used in reverse. After removing it, always check whether the bracketed leftover factors further — taking out the GCF is the beginning of complete factoring, not the end of it.