Factor the greatest common monomial factor

The greatest common monomial factor of a polynomial is the product of the numerical greatest common divisor of its coefficients and each variable raised to the smallest exponent appearing in every term, treating an absent variable as exponent zero. Factoring it out uses the distributive property to express the polynomial as a monomial times a polynomial, with no remaining shared monomial factor; the scope is limited to integer-coefficient monomials, binomials, and trinomials, not rational functions or more abstract coefficient systems.

Detailed Explanation: Factor the greatest common monomial factor

To factor the greatest common monomial factor, find what every term has in common, then use the distributive property.

Example

Factor:

18x^3y-24x^2y^2+30xy

1. Find the greatest common divisor of the coefficients.

The coefficients are $18$ , $24$ , and $30$ .

\gcd(18,24,30)=6

So, $6$ is part of the greatest common factor.

2. Find the variables common to every term.

The powers of $x$ are $x^3$ , $x^2$ , and $x^1$ . Use the smallest exponent: $x$ .
The powers of $y$ are $y^1$ , $y^2$ , and $y^1$ . Use the smallest exponent: $y$ .

Therefore, the greatest common monomial factor is

6xy

3. Divide each term by $6xy$ .

\frac{18x^3y}{6xy}=3x^2

\frac{-24x^2y^2}{6xy}=-4xy

\frac{30xy}{6xy}=5

4. Write the factored expression.

Place $6xy$ outside parentheses and write the results inside:

\boxed{18x^3y-24x^2y^2+30xy=6xy(3x^2-4xy+5)}

You can check by distributing $6xy$ :

6xy(3x^2-4xy+5)=18x^3y-24x^2y^2+30xy

The expression inside the parentheses has no monomial factor shared by all three terms, so the factoring is complete.

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