How To Factor A Common Factor

Factoring is a fundamental skill in algebra, allowing you to simplify expressions and solve equations more effectively. One of the most basic and crucial factoring techniques is factoring out a common factor. This process involves identifying the greatest common factor (GCF) that is present in each term of an expression and then factoring it out, leaving behind a simplified expression. Mastering this technique not only streamlines algebraic manipulations but also lays the groundwork for more advanced factoring methods.

What is Factoring and Why is it Important?

Factoring, in essence, is the reverse of expanding or distributing. When you distribute, you multiply a term across an expression in parentheses. For instance:

3(x + 2) = 3x + 6

Here, we've distributed the 3 across (x + 2). Factoring aims to undo this process. Given 3x + 6, factoring involves finding the common element (3) and rewriting the expression as:

3(x + 2)

Why is this important?

• Simplification: Factoring simplifies complex expressions, making them easier to work with.
• Solving Equations: Factoring is essential for solving polynomial equations. By setting a factored expression equal to zero, you can easily find the roots of the equation.
• Understanding Relationships: Factoring can reveal underlying relationships between terms in an expression, offering insights into its structure.
• Calculus: Factoring is crucial in calculus for simplifying expressions before differentiation or integration.

Understanding the Greatest Common Factor (GCF)

The foundation of factoring out a common factor lies in identifying the Greatest Common Factor (GCF). The GCF is the largest number or variable that divides evenly into all terms of an expression.

Finding the GCF for Numbers:

To find the GCF of a set of numbers, you can use a couple of methods:

1. Listing Factors: List all the factors of each number and identify the largest factor they have in common.

   • Example: Find the GCF of 12 and 18.

     • Factors of 12: 1, 2, 3, 4, 6, 12
     • Factors of 18: 1, 2, 3, 6, 9, 18

     The GCF of 12 and 18 is 6.

2. Prime Factorization: Break down each number into its prime factors and identify the common prime factors, multiplying them together to find the GCF.

   • Example: Find the GCF of 24 and 36.

     • Prime factorization of 24: 2 x 2 x 2 x 3 (2³ x 3)
     • Prime factorization of 36: 2 x 2 x 3 x 3 (2² x 3²)

     The common prime factors are 2 x 2 x 3. So, the GCF is 2 x 2 x 3 = 12.

Finding the GCF for Variables:

For variables, the GCF is the variable with the smallest exponent present in all terms.

• Example: Find the GCF of x³, x², and x⁵.

  The smallest exponent is 2. Therefore, the GCF is x².

Finding the GCF for Terms with Numbers and Variables:

Combine the methods for numbers and variables.

• Example: Find the GCF of 12x²y and 18xy³.

  • GCF of 12 and 18: 6
  • GCF of x² and x: x
  • GCF of y and y³: y

  Therefore, the GCF of 12x²y and 18xy³ is 6xy.

Steps to Factor Out a Common Factor

Now that you understand how to find the GCF, let's outline the steps to factor out a common factor from an expression:

1. Identify the GCF: Determine the greatest common factor of all the terms in the expression.
2. Divide Each Term by the GCF: Divide each term in the original expression by the GCF you found in step 1.
3. Write the Factored Expression: Write the GCF outside a set of parentheses. Inside the parentheses, write the results of the division from step 2, separated by the original signs.

Let's illustrate this with some examples:

Example 1: Factoring out a numerical GCF

Factor: 15x + 25

1. Identify the GCF: The GCF of 15 and 25 is 5.
2. Divide Each Term by the GCF:
   • 15x / 5 = 3x
   • 25 / 5 = 5
3. Write the Factored Expression: 5(3x + 5)

Therefore, 15x + 25 factored is 5(3x + 5).

Example 2: Factoring out a variable GCF

Factor: 4y³ - 8y² + 12y

1. Identify the GCF: The GCF of y³, y², and y is y (the smallest exponent).
2. Divide Each Term by the GCF:
   • 4y³ / y = 4y²
   • -8y² / y = -8y
   • 12y / y = 12
3. Write the Factored Expression: y(4y² - 8y + 12)

Therefore, 4y³ - 8y² + 12y factored is y(4y² - 8y + 12). Notice we could factor this further! The GCF of 4, -8 and 12 is 4. Thus, we can rewrite this as: 4y(y² - 2y + 3)

Example 3: Factoring out a combination of numerical and variable GCF

Factor: 18a²b³ + 24a³b² - 30a²b²

1. Identify the GCF:

   • GCF of 18, 24, and 30: 6
   • GCF of a², a³, and a²: a²
   • GCF of b³, b², and b²: b²

   Therefore, the GCF is 6a²b².

2. Divide Each Term by the GCF:

   • 18a²b³ / (6a²b²) = 3b
   • 24a³b² / (6a²b²) = 4a
   • -30a²b² / (6a²b²) = -5

3. Write the Factored Expression: 6a²b²(3b + 4a - 5)

Therefore, 18a²b³ + 24a³b² - 30a²b² factored is 6a²b²(3b + 4a - 5).

Example 4: Factoring out a Negative GCF

Sometimes, the leading coefficient of the expression is negative. In such cases, it's often helpful to factor out a negative GCF. This changes the signs of the terms inside the parentheses.

Factor: -4x + 8

1. Identify the GCF: The GCF of 4 and 8 is 4. Since the leading coefficient is negative, we factor out -4.
2. Divide Each Term by the GCF:
   • -4x / -4 = x
   • 8 / -4 = -2
3. Write the Factored Expression: -4(x - 2)

Therefore, -4x + 8 factored is -4(x - 2). Notice that if you factored out +4 instead, you'd get 4(-x + 2), which is also correct, but -4(x-2) is generally preferred.

Example 5: Factoring from expressions with more complex terms

Factor: 12(a + b)² - 18(a + b)

1. Identify the GCF: Notice that the term (a + b) is common to both parts of the expression.
   • GCF of 12 and 18: 6
   • GCF of (a + b)² and (a + b): (a + b) Therefore, the GCF is 6(a + b)
2. Divide Each Term by the GCF:
   • 12(a + b)² / 6(a + b) = 2(a + b)
   • -18(a + b) / 6(a + b) = -3
3. Write the Factored Expression: 6(a + b)[2(a + b) - 3]

We can simplify the expression further by distributing the 2: 6(a + b)(2a + 2b - 3)

Checking Your Work:

The best way to ensure you've factored correctly is to distribute the GCF back into the parentheses. The result should match the original expression.

• Example: We factored 5(3x + 5) from 15x + 25.

  • Distribute: 5(3x + 5) = 15x + 25. This matches the original expression, so our factoring is correct.

Common Mistakes to Avoid

• Forgetting to divide every term by the GCF: Make sure you divide each term in the original expression by the GCF.
• Incorrectly identifying the GCF: Double-check that you've found the greatest common factor, not just a common factor.
• Incorrectly dividing variables: Remember the rules of exponents when dividing variables with exponents (subtract the exponents).
• Not checking your work: Always distribute the factored expression to ensure it matches the original.
• Stopping too early: Sometimes, after factoring out a common factor, the expression inside the parentheses can be factored further. Always check for this possibility!

More Complex Examples and Applications

Factoring out a common factor is often the first step in more complex factoring problems. Here are some scenarios where it's particularly useful:

1. Factoring by Grouping

Factoring by grouping is used when you have four or more terms and can't immediately identify a GCF for all terms. The steps involve:

1. Grouping Terms: Group the terms into pairs.
2. Factoring out a GCF from each pair: Factor out the GCF from each pair of terms.
3. Identifying a Common Binomial Factor: If the two resulting terms have a common binomial factor (an expression in parentheses), factor it out.

• Example: Factor x³ + 3x² + 2x + 6

  1. Group: (x³ + 3x²) + (2x + 6)
  2. Factor from each pair: x²(x + 3) + 2(x + 3)
  3. Factor out the common binomial: (x + 3)(x² + 2)

2. Simplifying Rational Expressions

Factoring is essential for simplifying rational expressions (fractions with polynomials in the numerator and denominator). You factor both the numerator and denominator and then cancel out any common factors.

• Example: Simplify (x² + 5x) / (x² + 2x)

  1. Factor the numerator: x(x + 5)
  2. Factor the denominator: x(x + 2)
  3. Cancel the common factor: x(x + 5) / x(x + 2) = (x + 5) / (x + 2)

3. Solving Equations

Factoring is a crucial step in solving polynomial equations. By setting the factored expression equal to zero, you can use the zero-product property (if a * b = 0, then a = 0 or b = 0) to find the solutions.

• Example: Solve x² - 3x = 0

  1. Factor: x(x - 3) = 0
  2. Apply the zero-product property: x = 0 or x - 3 = 0
  3. Solve for x: x = 0 or x = 3

Tips for Mastering Factoring

• Practice Regularly: The more you practice, the more comfortable you'll become with identifying GCFs and factoring expressions.
• Start with Simple Examples: Begin with basic examples and gradually work your way up to more complex problems.
• Review Basic Algebra Skills: A solid understanding of the distributive property, exponents, and combining like terms is essential for factoring.
• Use Online Resources: Numerous websites and videos offer explanations and practice problems for factoring.
• Don't Be Afraid to Ask for Help: If you're struggling, ask your teacher, tutor, or classmates for assistance.

Conclusion

Factoring out a common factor is a foundational skill in algebra. Mastering this technique allows you to simplify expressions, solve equations, and gain a deeper understanding of mathematical relationships. By understanding how to identify the greatest common factor and following the steps outlined in this article, you'll be well on your way to mastering this essential algebraic skill. Remember to practice regularly and check your work to ensure accuracy. With consistent effort, you'll find that factoring becomes a powerful tool in your mathematical arsenal.