common monomial factor

Imagine every term of a polynomial carries the same ingredient — say each one has a factor of 3 and at least one x in it. The common monomial factor is the biggest single chunk made of numbers and variables that you can pull out of every term at once. Taking it out is almost always the very first step of factoring, the way you would simplify a fraction before doing anything fancier.

To find it, take the greatest common factor of the numerical coefficients and, for each variable, the smallest power that appears in any term. For 6x^3 + 9x^2, the number part is GCF(6, 9) = 3 and the variable part is x^2 (the smaller of x^3 and x^2), so the common monomial factor is 3x^2 and we write 6x^3 + 9x^2 = 3x^2(2x + 3).

After you remove this factor, the polynomial left inside the parentheses may still factor further — pulling out the monomial does not finish the job, it only clears the easy part. Always look for the common monomial factor first, because it shrinks the numbers and degrees you have to deal with afterward.