AI Tutor
If you multiply 3x(2x+5)3x(2x + 5), the distributive property gives you 6x2+15x6x^2 + 15x. But how do you run that movie backward to recover the original parts?
This reverse process is called factoring, and your first tool is pulling out the greatest common factor (or GCF) — the largest monomial that divides evenly into every term.
How do we systematically identify the largest shared chunk of numbers and variables across all terms?

Finding the Monomial GCF

A monomial is an algebraic expression with just one term, made of numbers and variables multiplied together (like 6x26x^2 or −4y-4y).
To find the GCF of multiple monomials, find the greatest common factor of the numerical coefficients (the numbers in front), then attach each shared variable raised to its lowest exponent.
Why do we always choose the lowest exponent? Because x3x^3 is x⋅x⋅xx \cdot x \cdot x and x2x^2 is x⋅xx \cdot x, meaning they only share two factors of xx in common.
Once we know the GCF is 6x26x^2, how do we determine what gets left behind inside the parentheses?

Factoring Out the GCF

We place the GCF outside parentheses, then divide each original term by the GCF to find the terms inside.
For the polynomial 12x3−18x212x^3 - 18x^2, dividing gives 12x36x2=2x\frac{12x^3}{6x^2} = 2x and −18x26x2=−3\frac{-18x^2}{6x^2} = -3, leaving us with the factored form 6x2(2x−3)6x^2(2x - 3).