How Do You Factor Out the Common Factor of Each Polynomial?

To factor out the common factor of each polynomial, you identify the greatest common factor (GCF) shared by all terms and then rewrite the polynomial using the distributive property.

Here's a breakdown of the process:

1. Identify the Greatest Common Factor (GCF)

• Look at the coefficients (numbers in front of the variables): Find the largest number that divides evenly into all the coefficients.
• Look at the variables: Identify the variable(s) common to all terms. Determine the lowest exponent of each common variable that appears in all terms.
• Combine: The GCF is the product of the largest common numerical factor and the lowest power of the common variable(s).

2. Apply the Distributive Property in Reverse

• Divide each term in the polynomial by the GCF you found.
• Rewrite the polynomial as the GCF multiplied by the expression resulting from the division in the previous step. This uses the distributive property in reverse: a*b + a*c = a*(b + c)

Example 1

Factor out the common factor of the polynomial: 6x^2 + 9x

1. Identify the GCF:

   • Coefficients: The largest number that divides evenly into both 6 and 9 is 3.
   • Variables: Both terms have x. The lowest exponent of x is 1 (in 9x).
   • GCF: 3x

2. Apply the distributive property in reverse:

   • Divide each term by 3x:
     • 6x^2 / 3x = 2x
     • 9x / 3x = 3
   • Rewrite the polynomial: 6x^2 + 9x = 3x(2x + 3)

Example 2

Factor out the common factor of the polynomial: 12a^3b^2 - 18a^2b^4 + 30a^4b^3

1. Identify the GCF:

   • Coefficients: The largest number that divides evenly into 12, -18, and 30 is 6.
   • Variables: All terms have a and b. The lowest exponent of a is 2 and of b is 2.
   • GCF: 6a^2b^2

2. Apply the distributive property in reverse:

   • Divide each term by 6a^2b^2:
     • 12a^3b^2 / 6a^2b^2 = 2a
     • -18a^2b^4 / 6a^2b^2 = -3b^2
     • 30a^4b^3 / 6a^2b^2 = 5a^2b
   • Rewrite the polynomial: 12a^3b^2 - 18a^2b^4 + 30a^4b^3 = 6a^2b^2(2a - 3b^2 + 5a^2b)

Summary

Factoring out the common factor involves finding the GCF of all terms in a polynomial and then using the distributive property in reverse to rewrite the polynomial as the GCF multiplied by a new polynomial expression. This simplifies the original polynomial and can be useful for solving equations and simplifying expressions.