How do you factor a variable with an exponent?

Factoring a variable with an exponent involves finding the greatest common factor (GCF) and then dividing the original expression by that GCF. The result expresses the original term as a product.

Factoring Out Variables with Exponents: A Detailed Guide

Factoring out a variable with an exponent is a fundamental algebraic skill. It's the reverse process of distribution and is used to simplify expressions and solve equations. Here's a comprehensive breakdown:

1. Identify the Greatest Common Factor (GCF):

   • When factoring an expression containing terms with exponents, the GCF is determined by the variable raised to the smallest exponent present in those terms.
   • If there are coefficients (numbers) in front of the variables, find the GCF of those coefficients as well.
   • According to the provided reference: "To do this, take the greatest common factor of the numbers and the smallest exponent of each variable."

2. Factor Out the GCF:

   • Rewrite each term in the expression as a product of the GCF and another factor.
   • This step visually separates the common element that you'll be factoring out.

3. Divide the original expression by the greatest common factor:

   • "To do this, divide the coefficients, and subtract the exponents of the variables."
   • When you divide variables with exponents, you subtract the exponents (using the rule x^m / xⁿ = x^m-n).

4. Write the Factored Expression:

   • The factored expression consists of the GCF outside of parentheses, followed by the expression remaining after you've divided each term by the GCF inside the parentheses.

Examples

Let's illustrate with examples:

Example 1: Factoring a Simple Expression

Factor the expression: 5x³ + 10x²

1. Identify the GCF:

   • The GCF of the coefficients (5 and 10) is 5.
   • The smallest exponent of x is 2 (x²).
   • Therefore, the GCF is 5x².

2. Factor out the GCF:

   • 5x³ + 10x² = 5x²(x) + 5x²(2)

3. Write the Factored Expression:

   • 5x²(x + 2)

Example 2: Factoring a More Complex Expression

Factor the expression: 12a⁴b² - 18a²b³

1. Identify the GCF:

   • The GCF of the coefficients (12 and 18) is 6.
   • The smallest exponent of 'a' is 2 (a²).
   • The smallest exponent of 'b' is 2 (b²).
   • Therefore, the GCF is 6a²b².

2. Factor out the GCF:

   • 12a⁴b² - 18a²b³ = 6a²b²(2a²) - 6a²b²(3b)

3. Write the Factored Expression:

   • 6a²b²(2a² - 3b)

Tips for Factoring

• Always look for a GCF first. This simplifies the factoring process.
• Double-check your work. Distribute the GCF back into the parentheses to make sure you arrive at the original expression.
• Practice regularly. Factoring becomes easier with practice.