Factoring 8c² - 2c - 3: A Step-by-Step Guide

Factoring quadratic expressions can seem daunting, but with a systematic approach, it becomes a manageable task. This guide will walk you through the process of factoring the quadratic expression 8c² - 2c - 3. We'll break down each step, providing clear explanations and helpful tips along the way. So, if you're ready to master factoring, let's dive in!

Understanding the Basics of Factoring Quadratics

Before we jump into the specifics of factoring 8c² - 2c - 3, let's quickly review the basics of factoring quadratic expressions. A quadratic expression is a polynomial of degree two, generally written in the form ax² + bx + c, where a, b, and c are constants. Factoring a quadratic expression means rewriting it as a product of two binomials. For example, the quadratic expression x² + 5x + 6 can be factored as (x + 2)(x + 3).

In our case, the expression is 8c² - 2c - 3. Here, a = 8, b = -2, and c = -3. Factoring this expression involves finding two binomials that, when multiplied together, result in 8c² - 2c - 3. Several methods can be used for factoring quadratics, but we'll focus on the AC method, which is particularly useful when the coefficient of the squared term (in this case, 8) is not 1.

The AC Method Explained

The AC method involves the following steps:

1. Multiply a and c: Calculate the product of the coefficient of the squared term (a) and the constant term (c). In our example, a = 8 and c = -3, so a × c = 8 × (-3) = -24.

2. Find two factors of ac that add up to b: We need to find two numbers that multiply to -24 and add up to b, which is -2. Let's list the factor pairs of -24 and see which pair fits the bill:

   • 1 and -24 (sum: -23)
   • -1 and 24 (sum: 23)
   • 2 and -12 (sum: -10)
   • -2 and 12 (sum: 10)
   • 3 and -8 (sum: -5)
   • -3 and 8 (sum: 5)
   • 4 and -6 (sum: -2)
   • -4 and 6 (sum: 2)

   The pair 4 and -6 satisfies our condition, as 4 × (-6) = -24 and 4 + (-6) = -2.

3. Rewrite the middle term: Replace the middle term (-2c) with the sum of the two terms using the factors we just found (4c and -6c). So, 8c² - 2c - 3 becomes 8c² + 4c - 6c - 3.

4. Factor by grouping: Group the first two terms and the last two terms and factor out the greatest common factor (GCF) from each group:

   • From the first group (8c² + 4c), the GCF is 4c. Factoring out 4c gives us 4c(2c + 1).
   • From the second group (-6c - 3), the GCF is -3. Factoring out -3 gives us -3(2c + 1). Now we have 4c(2c + 1) - 3(2c + 1).

5. Factor out the common binomial: Notice that both terms now have a common binomial factor, (2c + 1). Factor this out to get the final factored form: (2c + 1)(4c - 3).

Step-by-Step Factoring of 8c² - 2c - 3

Let's walk through the factoring process again, step by step, to solidify your understanding.

Step 1: Multiply a and c

As we discussed earlier, a = 8 and c = -3. Therefore, a × c = 8 × (-3) = -24.

Step 2: Find Two Factors of -24 That Add Up to -2

We need to find two numbers that multiply to -24 and add up to -2. We already identified these factors as 4 and -6.

Step 3: Rewrite the Middle Term

Replace -2c with 4c - 6c in the original expression:

8c² - 2c - 3 becomes 8c² + 4c - 6c - 3.

Step 4: Factor by Grouping

Group the first two terms and the last two terms:

(8c² + 4c) + (-6c - 3)

Factor out the GCF from each group:

• From 8c² + 4c, the GCF is 4c. Factoring out 4c gives us 4c(2c + 1).
• From -6c - 3, the GCF is -3. Factoring out -3 gives us -3(2c + 1).

So we have: 4c(2c + 1) - 3(2c + 1).

Step 5: Factor Out the Common Binomial

Both terms have the common binomial factor (2c + 1). Factor this out:

(2c + 1)(4c - 3)

Therefore, the factored form of 8c² - 2c - 3 is (2c + 1)(4c - 3).

Verifying the Factored Form

To ensure our factoring is correct, we can multiply the binomials (2c + 1) and (4c - 3) together using the FOIL method (First, Outer, Inner, Last) and see if we get back the original expression:

• First: 2c × 4c = 8c²
• Outer: 2c × -3 = -6c
• Inner: 1 × 4c = 4c
• Last: 1 × -3 = -3

Adding these terms together, we get:

8c² - 6c + 4c - 3 = 8c² - 2c - 3

This matches the original expression, so our factoring is correct.

Tips for Mastering Factoring

• Practice Regularly: The more you practice factoring, the more comfortable you'll become with the process. Try factoring a variety of quadratic expressions with different coefficients and constants.
• Understand the Methods: Familiarize yourself with different factoring methods, such as the AC method, grouping, and recognizing special patterns (like the difference of squares). Choose the method that works best for you and the specific problem.
• Check Your Work: Always multiply your factored binomials back together to ensure you arrive at the original expression. This is a crucial step in verifying the correctness of your factoring.
• Look for GCF First: Before applying any other factoring method, always check if there's a greatest common factor (GCF) that can be factored out from all terms. This can simplify the expression and make it easier to factor.
• Don't Give Up: Factoring can be challenging at first, but with persistence and practice, you'll improve your skills and develop a strong understanding of the concepts. If you get stuck, review the steps, try a different method, or seek help from a teacher or tutor.

Common Mistakes to Avoid

• Incorrectly Identifying Factors: Double-check the factors you've identified to ensure they multiply to the correct product (ac) and add up to the correct sum (b).
• Sign Errors: Pay close attention to the signs of the factors and terms. A small sign error can lead to an incorrect factored form.
• Forgetting to Factor Completely: Ensure that you've factored the expression as much as possible. Sometimes, after applying a factoring method, you may need to factor one of the resulting binomials further.
• Not Checking Your Work: Always verify your factored form by multiplying the binomials back together. This will help you catch any errors and ensure your solution is correct.

Conclusion

Factoring the quadratic expression 8c² - 2c - 3 involves a systematic approach using the AC method, which includes finding factors, rewriting terms, and grouping. By following the steps outlined in this guide and practicing regularly, you can confidently factor a wide range of quadratic expressions. Remember to verify your work and avoid common mistakes to ensure accuracy. Happy factoring!

For further learning and practice, consider exploring resources on websites like Khan Academy's Algebra section to deepen your understanding of factoring and other algebraic concepts.