askvity

How do you factor exponents with variables?

Published in Factoring Expressions 3 mins read

Factoring expressions with exponents and variables involves identifying the greatest common factor (GCF) of the terms and extracting it from the expression. This simplifies the expression and can be useful for solving equations or simplifying further calculations.

Factoring Explained

The core concept is to find the GCF, which consists of the largest numerical factor and the smallest exponent of each common variable.

  • Greatest Common Factor (GCF): The largest factor that divides evenly into all terms of the expression.
  • Variables and Exponents: When variables are involved, identify the common variables in all terms and select the smallest exponent for each.

Steps to Factor Exponents with Variables

Here's a step-by-step guide with examples:

  1. Identify the GCF:
    • Find the largest number that divides evenly into all the coefficients (the numbers in front of the variables).
    • Identify the variables present in all terms.
    • For each common variable, choose the smallest exponent.
  2. Extract the GCF: Write the GCF outside a set of parentheses.
  3. Divide by the GCF: Divide each term in the original expression by the GCF. This involves dividing the coefficients and subtracting the exponents of the variables.
  4. Write the Result: Place the results of the division inside the parentheses.

Examples

Let's illustrate with a few examples:

Example 1

Factor the expression: 12x^3 + 18x^2

  1. Identify the GCF:
    • The greatest common factor of 12 and 18 is 6.
    • The common variable is x.
    • The smallest exponent of x is 2 (from x^2).
    • Therefore, the GCF is 6x^2.
  2. Extract the GCF: 6x^2( )
  3. Divide by the GCF:
    • 12x^3 / 6x^2 = 2x
    • 18x^2 / 6x^2 = 3
  4. Write the Result: 6x^2(2x + 3)

So, the factored form of 12x^3 + 18x^2 is 6x^2(2x + 3).

Example 2

Factor the expression: 4a^5b^2 - 8a^3b^3 + 12a^2b^4

  1. Identify the GCF:
    • The greatest common factor of 4, 8, and 12 is 4.
    • The common variables are a and b.
    • The smallest exponent of a is 2 (from a^2).
    • The smallest exponent of b is 2 (from b^2).
    • Therefore, the GCF is 4a^2b^2.
  2. Extract the GCF: 4a^2b^2( )
  3. Divide by the GCF:
    • 4a^5b^2 / 4a^2b^2 = a^3
    • -8a^3b^3 / 4a^2b^2 = -2ab
    • 12a^2b^4 / 4a^2b^2 = 3b^2
  4. Write the Result: 4a^2b^2(a^3 - 2ab + 3b^2)

Thus, the factored form of 4a^5b^2 - 8a^3b^3 + 12a^2b^4 is 4a^2b^2(a^3 - 2ab + 3b^2).

Summary of Steps (based on the reference)

Step Description Example (Using 12x^3 + 18x^2)
1. Find the GCF Take the greatest common factor of the numbers and the smallest exponent of each variable. GCF of 12 and 18 is 6. Smallest exponent of x is 2. GCF = 6x^2
2. Divide by the GCF Divide the original expression by the greatest common factor. 12x^3 / 6x^2 = 2x; 18x^2 / 6x^2 = 3
3. Write Factored Form To do this, divide the coefficients, and subtract the exponents of the variables and write in factored form. 6x^2(2x + 3)

By following these steps, you can efficiently factor expressions containing exponents and variables.

Related Articles