Finding Factors One Factor Of 12x² – 41x – 15 A Step-by-Step Guide

by Scholario Team 67 views

Hey guys! Ever stumbled upon a quadratic expression and felt like you're staring at an alien language? Well, you're not alone! Today, we're going to break down a common type of math problem: factoring quadratic expressions. Specifically, we'll tackle the expression 12x² – 41x – 15. Don't worry, we'll take it step by step, and by the end of this guide, you'll be factoring like a pro. So, let's dive in and figure out one of the factors of this expression. Remember, understanding these concepts is crucial, not just for exams but also for building a solid foundation in mathematics. Think of it as unlocking a new level in your math skills!

Understanding Quadratic Expressions

Before we jump into solving, let's make sure we're all on the same page. A quadratic expression is essentially a polynomial equation with the highest power of the variable being 2. The general form looks like this: ax² + bx + c, where a, b, and c are constants, and x is the variable. In our case, we have 12x² – 41x – 15, so a is 12, b is -41, and c is -15. These expressions pop up everywhere in math and science, from calculating the trajectory of a ball to designing bridges. Understanding them is like having a superpower in the world of problem-solving. Factoring, in simple terms, is like reverse engineering. We're trying to find the two expressions that, when multiplied together, give us our original quadratic expression. It's like finding the ingredients that make up a cake. If you grasp the basics, the more complex stuff becomes way easier to handle. So, keep practicing, and you'll become a quadratic expression master in no time!

The Factoring Process: A Detailed Walkthrough

Okay, let's get down to business! Factoring 12x² – 41x – 15 might seem daunting at first, but we'll use a systematic approach to make it manageable. Here's the breakdown:

1. The AC Method:

The AC method is our go-to technique for factoring quadratics when the coefficient of x² (our a value) isn't 1. It's a bit like a puzzle, but once you get the hang of it, it's super effective. First, we multiply a and c. In our case, that's 12 * (-15) = -180. This number is crucial because it sets the stage for the next step. We need to find two numbers that multiply to -180 and, at the same time, add up to b (-41). This might seem tricky, but there are ways to make it easier, which we'll explore next.

2. Finding the Right Pair of Numbers:

This is where the detective work comes in! We need two numbers that multiply to -180 and add up to -41. Let's think about factors of 180. We could have 1 and 180, 2 and 90, 3 and 60, and so on. Since we need a negative product, one number must be positive, and the other negative. And because they need to add up to -41, the larger number should be negative. After some trial and error (and maybe a little calculator help!), we find that -45 and 4 fit the bill perfectly. Why? Because -45 * 4 = -180 and -45 + 4 = -41. This step is the heart of the AC method. Once you nail this, the rest is relatively smooth sailing.

3. Rewriting the Middle Term:

Now, we take our original expression, 12x² – 41x – 15, and rewrite the middle term (-41x) using the numbers we just found (-45 and 4). So, -41x becomes -45x + 4x. Our expression now looks like this: 12x² - 45x + 4x - 15. This might seem like we're making things more complicated, but trust me, it's a necessary step to factoring by grouping, which is our next move. Think of it as rearranging the pieces of a puzzle to fit them together better.

4. Factoring by Grouping:

This technique is all about finding common factors within groups of terms. We've split our four-term expression (12x² - 45x + 4x - 15) into two pairs: (12x² - 45x) and (4x - 15). Now, we factor out the greatest common factor (GCF) from each pair. For the first pair, the GCF is 3x, so we get 3x(4x - 15). For the second pair, the GCF is 1 (since there's no other common factor), so we get 1(4x - 15). Notice something cool? Both groups now have a common factor of (4x - 15). This is a great sign! It means we're on the right track. Factoring by grouping is a bit like sorting your socks – you pair them up and then look for what they have in common.

5. Final Factorization:

We're almost there! Since both groups have the factor (4x - 15), we can factor it out. We're left with: (4x - 15)(3x + 1). And that's it! We've factored the quadratic expression. This is our factored form. It's like the final, polished product after all the hard work. Now, to answer the original question, one of the factors of 12x² – 41x – 15 is (4x - 15) or (3x + 1). You've successfully navigated the factoring process!

Checking Your Answer: The FOIL Method

Want to be 100% sure you've nailed it? Let's double-check our answer using the FOIL method. FOIL stands for First, Outer, Inner, Last, and it's a way to multiply two binomials (expressions with two terms) together. It's like the ultimate proofreading technique for factoring. We'll multiply (4x - 15) and (3x + 1) using FOIL:

  • First: Multiply the first terms in each binomial: 4x * 3x = 12x²
  • Outer: Multiply the outer terms: 4x * 1 = 4x
  • Inner: Multiply the inner terms: -15 * 3x = -45x
  • Last: Multiply the last terms: -15 * 1 = -15

Now, let's add these results together: 12x² + 4x - 45x - 15. Combine the like terms (4x and -45x), and we get 12x² - 41x - 15. Guess what? That's our original expression! So, we know our factoring is correct. The FOIL method is like having a secret decoder ring to make sure your work is spot on. It's an essential tool in your math arsenal!

Common Mistakes to Avoid

Factoring can be tricky, and even the best of us make mistakes sometimes. Here are a few common pitfalls to watch out for:

  • Sign Errors: Keep a close eye on your signs, especially when dealing with negative numbers. A small sign error can throw off your entire solution. Always double-check that you've correctly identified positive and negative factors.
  • Incorrect Multiplication/Addition: Make sure the numbers you choose multiply to the correct ac value and add up to the correct b value. This is the heart of the AC method, so precision is key.
  • Forgetting to Factor Completely: Sometimes, after factoring, you might have a factor that can be factored further. Always check if your factors can be simplified more.
  • Skipping Steps: It's tempting to rush through the process, but skipping steps can lead to errors. Take your time, write out each step clearly, and double-check your work.

By being aware of these common mistakes, you can avoid them and boost your factoring accuracy. Remember, practice makes perfect! The more you factor, the more these steps will become second nature.

Practice Problems

Alright, time to put your skills to the test! Here are a few practice problems to help you master factoring quadratic expressions:

  1. 6x² + 13x + 6
  2. 8x² - 10x - 3
  3. 9x² + 12x + 4

Try factoring these on your own, using the steps we've discussed. Remember, the key is to practice consistently. The more you work through these problems, the more confident and proficient you'll become. Don't be afraid to make mistakes – they're part of the learning process. And if you get stuck, revisit the steps we covered earlier or ask for help. Happy factoring!

Conclusion

So, there you have it, guys! We've walked through the process of finding one of the factors of 12x² – 41x – 15 step by step. We started with understanding quadratic expressions, used the AC method, factored by grouping, and even checked our answer with the FOIL method. Factoring can be challenging, but with practice and a solid understanding of the steps, you can conquer any quadratic expression that comes your way. Remember, math is like a muscle – the more you exercise it, the stronger it gets. Keep practicing, stay curious, and you'll be amazed at what you can achieve! Now, go forth and factor like a champ!