Definition: The greatest common factor of an expression is the largest number and highest power of each variable that divide every one of its terms evenly.
How do you find it?
Find the greatest common factor of the coefficients. Then, for each variable that appears in every term, take the smallest exponent it has. Multiply those pieces.
How do you pull it out?
Write the factor outside parentheses and divide each term by it to get what stays inside.
Example. Factor 18x4y – 27x2y3 + 45x3y2. Coefficients 18, 27, 45 share 9. The smallest power of x is 2 and of y is 1. So the factor is 9x2y. Dividing: 18x4y becomes 2x2, 27x2y3 becomes 3y2, and 45x3y2 becomes 5xy. The result is 9x2y(2x2 – 3y2 + 5xy).
What is easy to miss?
If a term equals the factor exactly, the leftover is 1, not nothing: 6x + 6 factors as 6(x + 1). Check your answer by distributing. A variable missing from even one term cannot be part of the common factor.