Factor a common monomial factor

Factoring a common monomial means rewriting a polynomial as a product by identifying a monomial that divides every term, typically using the greatest common numerical factor and each variable raised to the smallest exponent appearing in all terms. The process reverses the distributive property, preserves equality, and clarifies polynomial structure for later work with simplifying expressions, solving equations, and factoring; it does not include non-monomial factoring methods such as grouping or quadratic-factor techniques.

Detailed Explanation: Factor a common monomial factor

To factor out a common monomial, find a monomial that divides every term. Use:

The greatest common factor of the numerical coefficients.
Each variable with the smallest exponent appearing in every term.

Then divide each term by the common monomial and write the result in parentheses.

Example

Factor:

12x^3y-18x^2y^2+6xy

Step 1: Find the greatest common numerical factor.

The coefficients are $12$ , $18$ , and $6$ . Their greatest common factor is $6$ .

Step 2: Find the common variables.

The smallest power of $x$ is $x^1$ , so include $x$ .
The smallest power of $y$ is $y^1$ , so include $y$ .

The common monomial is therefore:

6xy

Step 3: Divide each term by $6xy$ .

12x^3y\div 6xy=2x^2

-18x^2y^2\div 6xy=-3xy

6xy\div 6xy=1

Step 4: Write the factored expression.

12x^3y-18x^2y^2+6xy=6xy(2x^2-3xy+1)

You can check by distributing $6xy$ back into the parentheses:

6xy(2x^2-3xy+1)=12x^3y-18x^2y^2+6xy

So the factored form is:

\boxed{6xy(2x^2-3xy+1)}

Learn by doing: Factor a common monomial factor

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Algebraic Functions - Remove Different Variable From Bracketed Terms

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