Skill: 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.