How do you factor the greatest common factor out of Monomials?

To factor the greatest common factor (GCF) out of monomials, you identify the largest factor that divides evenly into all terms and then rewrite the expression with the GCF factored out.

Steps to Factor the GCF out of Monomials:

1. Find the GCF of the Coefficients: Determine the largest number that divides evenly into the coefficients of all the monomials.

2. Find the GCF of the Variables: For each variable, identify the lowest exponent that appears in any of the monomials. The GCF for that variable will be the variable raised to that lowest exponent. If a variable doesn't appear in all monomials, it's not part of the variable GCF.

3. Combine the GCFs: Multiply the GCF of the coefficients and the GCF of the variables to get the overall GCF of the monomials.

4. Factor out the GCF: Divide each monomial by the GCF and write the result inside parentheses. The GCF will be written outside the parentheses. This is the factored form.

Example:

Let's say you want to factor the GCF out of the following monomials:

12x³y² + 18x²y³ - 30x⁴y

1. GCF of Coefficients: The GCF of 12, 18, and 30 is 6.

2. GCF of Variables:

   • For x, the lowest exponent is 2 (x²).
   • For y, the lowest exponent is 1 (y¹ or simply y).

3. Overall GCF: 6x²y

4. Factor out the GCF:

   12x³y² + 18x²y³ - 30x⁴y = 6x²y(2xy + 3y² - 5x²)

In Summary:

Factoring the greatest common factor out of monomials involves finding the largest factor that divides evenly into all terms (both coefficients and variables) and then rewriting the expression as a product of the GCF and the remaining terms.