Factor The Greatest Common Factor From The Polynomial

Factoring the greatest common factor (GCF) from a polynomial is a fundamental skill in algebra. It's the process of identifying and extracting the largest expression that divides evenly into each term of the polynomial, simplifying the expression and paving the way for more advanced factoring techniques. This article will guide you through the process, providing clear explanations and examples to help you master this essential skill.

Understanding the Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest factor that two or more numbers or expressions share. When factoring polynomials, the GCF can be a number, a variable, or a combination of both. Identifying the GCF is the first step in simplifying many algebraic expressions.

• Numerical GCF: The largest number that divides evenly into all the coefficients in the polynomial.
• Variable GCF: The variable with the lowest exponent that appears in all terms of the polynomial.

Steps to Factor the Greatest Common Factor

Factoring the GCF involves a systematic approach:

1. Identify the GCF: Determine the greatest common factor of the coefficients and variables in the polynomial.
2. Write the polynomial as a product: Express each term of the polynomial as a product of the GCF and its remaining factor.
3. Factor out the GCF: Write the GCF outside of parentheses, followed by the sum (or difference) of the remaining factors inside the parentheses.
4. Verify the result: Distribute the GCF back into the parentheses to ensure you obtain the original polynomial.

Detailed Walkthrough with Examples

Let's illustrate the factoring process with several examples:

Example 1: Factoring a Simple Polynomial

Polynomial: 6x + 12

1. Identify the GCF:
   • Numerical GCF: The largest number that divides both 6 and 12 is 6.
   • Variable GCF: There is no variable GCF because the second term, 12, doesn't contain 'x'.
   • Therefore, the GCF is 6.
2. Write the polynomial as a product:
   • 6x = 6 * x
   • 12 = 6 * 2
3. Factor out the GCF:
   • 6x + 12 = 6(x + 2)
4. Verify the result:
   • 6(x + 2) = 6 * x + 6 * 2 = 6x + 12 (Original polynomial)

Example 2: Factoring with Variable GCF

Polynomial: 5x² - 10x

1. Identify the GCF:
   • Numerical GCF: The largest number that divides both 5 and 10 is 5.
   • Variable GCF: Both terms contain 'x'. The lowest exponent of 'x' is x¹ (or simply x).
   • Therefore, the GCF is 5x.
2. Write the polynomial as a product:
   • 5x² = 5x * x
   • -10x = 5x * (-2)
3. Factor out the GCF:
   • 5x² - 10x = 5x(x - 2)
4. Verify the result:
   • 5x(x - 2) = 5x * x - 5x * 2 = 5x² - 10x (Original polynomial)

Example 3: Factoring with Multiple Variables

Polynomial: 12a³b² + 18a²b³

1. Identify the GCF:
   • Numerical GCF: The largest number that divides both 12 and 18 is 6.
   • Variable GCF: Both terms contain 'a' and 'b'. The lowest exponent of 'a' is a² and the lowest exponent of 'b' is b².
   • Therefore, the GCF is 6a²b².
2. Write the polynomial as a product:
   • 12a³b² = 6a²b² * 2a
   • 18a²b³ = 6a²b² * 3b
3. Factor out the GCF:
   • 12a³b² + 18a²b³ = 6a²b²(2a + 3b)
4. Verify the result:
   • 6a²b²(2a + 3b) = 6a²b² * 2a + 6a²b² * 3b = 12a³b² + 18a²b³ (Original polynomial)

Example 4: Factoring with Negative Coefficients

Polynomial: -4x³ + 8x² - 12x

1. Identify the GCF:
   • Numerical GCF: You can factor out either 4 or -4. Factoring out the negative often makes the remaining polynomial easier to work with. Let's use -4.
   • Variable GCF: The lowest power of x present in each term is x.
   • Therefore, the GCF is -4x.
2. Write the polynomial as a product:
   • -4x³ = -4x * x²
   • 8x² = -4x * (-2x)
   • -12x = -4x * 3
3. Factor out the GCF:
   • -4x³ + 8x² - 12x = -4x(x² - 2x + 3)
4. Verify the result:
   • -4x(x² - 2x + 3) = -4x * x² - 4x * (-2x) - 4x * 3 = -4x³ + 8x² - 12x (Original polynomial)

Example 5: Factoring with More Complex Expressions

Polynomial: 9(x + 2)² - 6(x + 2)

1. Identify the GCF:
   • Numerical GCF: The largest number that divides both 9 and 6 is 3.
   • Expression GCF: Both terms contain the expression (x + 2). The lowest power of (x + 2) is (x + 2)¹, or simply (x + 2).
   • Therefore, the GCF is 3(x + 2).
2. Write the polynomial as a product:
   • 9(x + 2)² = 3(x + 2) * 3(x + 2)
   • -6(x + 2) = 3(x + 2) * (-2)
3. Factor out the GCF:
   • 9(x + 2)² - 6(x + 2) = 3(x + 2)[3(x + 2) - 2]
4. Simplify the expression inside the brackets:
   • 3(x + 2) - 2 = 3x + 6 - 2 = 3x + 4
5. Final factored form:
   • 9(x + 2)² - 6(x + 2) = 3(x + 2)(3x + 4)
6. Verify the result (this is a bit more involved, but important):
   • 3(x + 2)(3x + 4) = 3(3x² + 4x + 6x + 8) = 3(3x² + 10x + 8) = 9x² + 30x + 24
   • Now expand the original expression: 9(x² + 4x + 4) - 6(x + 2) = 9x² + 36x + 36 - 6x - 12 = 9x² + 30x + 24
   • Both expressions match!

Common Mistakes to Avoid

• Missing a factor: Ensure you've identified the greatest common factor, not just a common factor.
• Incorrect exponents: When factoring out variables, use the lowest exponent present in all terms.
• Forgetting to distribute: Always multiply the GCF back into the parentheses to verify that you get the original polynomial.
• Sign errors: Pay close attention to signs, especially when factoring out a negative GCF.
• Stopping too early: After factoring out the GCF, check if the remaining polynomial inside the parentheses can be factored further.

The Importance of Factoring the GCF

Factoring the GCF is a crucial first step in many algebraic problems for several reasons:

• Simplification: It reduces the complexity of the polynomial, making it easier to work with.
• Solving Equations: It helps in solving polynomial equations by allowing you to use the zero-product property (if a*b = 0, then a = 0 or b = 0).
• Further Factoring: It often reveals a simpler polynomial that can be factored further using techniques like difference of squares, perfect square trinomials, or grouping.
• Calculus Applications: Factoring is fundamental in calculus for simplifying expressions before differentiation or integration.

Advanced Techniques and Considerations

• Factoring by Grouping: When a polynomial has four or more terms, factoring by grouping may be necessary after factoring out the GCF (if one exists across all terms).
• Nested Factoring: Sometimes, after factoring out the GCF, the remaining polynomial requires factoring again. Always check for this possibility.
• Prime Polynomials: If the only common factor among the terms is 1, the polynomial is considered prime and cannot be factored further (using GCF only).

Practice Problems

To solidify your understanding, try factoring the GCF from the following polynomials:

1. 15x³ + 25x²
2. 8a⁴b - 12a²b³
3. -6y⁵ + 18y³ - 24y
4. 14(p - 1)³ + 21(p - 1)
5. 4x²yz³ - 12xy²z + 8xyz²

Answers:

1. 5x²(3x + 5)
2. 4a²b(2a² - 3b²)
3. -6y(y⁴ - 3y² + 4)
4. 7(p - 1)[2(p - 1)² + 3] (which can be further simplified to 7(p-1)(2p² - 4p + 5))
5. 4xyz(xz² - 3y + 2z)

Conclusion

Factoring the greatest common factor is a foundational skill in algebra. By understanding the concept of the GCF and following the steps outlined in this article, you can confidently simplify polynomials and prepare for more advanced factoring techniques. Remember to practice regularly and verify your results to build accuracy and fluency. Mastering this skill will significantly enhance your algebraic abilities and open doors to solving more complex mathematical problems.