Factoring Trinomials X^2 + 4x - 21 A Step-by-Step Solution

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Factoring trinomials might seem daunting at first, but trust me, guys, it's like solving a puzzle once you get the hang of it! Let's break down how to factor the trinomial x² + 4x - 21, explore the correct answer, and understand why the other options are incorrect. We'll make sure you're a factoring pro by the end of this article!

Understanding the Trinomial

Before we dive into factoring, let's quickly define what a trinomial actually is. A trinomial, simply put, is a polynomial expression that contains three terms. These terms are typically arranged in descending order of their exponents. In our case, x² + 4x - 21 is indeed a trinomial because it has three terms: x² (a quadratic term), 4x (a linear term), and -21 (a constant term). When we talk about factoring a trinomial, we're essentially trying to reverse the process of multiplication. Think of it like this: if you multiply two binomials (expressions with two terms), you often get a trinomial. Factoring is the process of finding those original binomials.

The Key to Factoring: Finding the Right Numbers

The key to successfully factoring a trinomial like x² + 4x - 21 lies in finding two numbers that meet specific criteria. We need two numbers that:

  1. Multiply to the constant term (-21 in this case).
  2. Add up to the coefficient of the linear term (4 in this case).

This might sound a bit abstract, so let's walk through how we can find these magic numbers for our trinomial. We need to consider the factors of -21. Remember, since the product is negative, one factor must be positive, and the other must be negative. Let's list out the possible pairs of factors:

  • 1 and -21
  • -1 and 21
  • 3 and -7
  • -3 and 7

Now, let's check which of these pairs adds up to 4 (the coefficient of our x term):

  • 1 + (-21) = -20 (Nope!)
  • -1 + 21 = 20 (Nope!)
  • 3 + (-7) = -4 (Close, but not quite!)
  • -3 + 7 = 4 (Bingo!)

So, the two numbers we're looking for are -3 and 7. This is the crucial step in factoring this type of trinomial. Once you've identified these numbers, the rest is relatively straightforward.

The Factoring Process: Putting It All Together

Now that we've found the numbers -3 and 7, we can express our trinomial x² + 4x - 21 as the product of two binomials. Here's how we do it:

  1. Start with two sets of parentheses: ( )( )
  2. Place 'x' as the first term in each binomial: (x )(x )
  3. Insert the two numbers we found (-3 and 7) into the binomials: (x - 3)(x + 7)

And there you have it! The factored form of x² + 4x - 21 is (x - 3)(x + 7). You can always double-check your work by multiplying the binomials back together using the FOIL method (First, Outer, Inner, Last) to see if you get the original trinomial. Let's quickly verify:

(x - 3)(x + 7) = x(x) + x(7) - 3(x) - 3(7) = x² + 7x - 3x - 21 = x² + 4x - 21

It checks out! So, our factored form is indeed correct.

Identifying the Correct Answer

Looking at the options provided, we can clearly see that option D. (x - 3)(x + 7) is the correct answer. This is the factored form we derived through our step-by-step process.

Why the Other Options Are Incorrect

Understanding why the other options are incorrect is just as important as knowing the correct answer. It helps you solidify your understanding of factoring and avoid common mistakes. Let's take a look at why options A, B, and C are wrong:

A. (x + 3)(x - 7)

If we multiply (x + 3)(x - 7) using the FOIL method, we get:

x² - 7x + 3x - 21 = x² - 4x - 21

Notice that the middle term is -4x, not +4x as in our original trinomial. This is because the signs are switched compared to our correct factors. This option is incorrect because the numbers 3 and -7 do multiply to -21, but they add up to -4, not 4.

B. (x - 3)(x - 7)

Multiplying (x - 3)(x - 7) gives us:

x² - 7x - 3x + 21 = x² - 10x + 21

In this case, the constant term is +21, not -21. This is a significant error. The factors -3 and -7 multiply to +21, which is not the constant term in our original trinomial. Also, the middle term is -10x, which is also incorrect. This option highlights the importance of getting the signs right.

C. (x + 3)(x + 7)

Expanding (x + 3)(x + 7) results in:

x² + 7x + 3x + 21 = x² + 10x + 21

Here, both the middle term (10x) and the constant term (+21) are incorrect. The numbers 3 and 7 add up to 10, not 4, and they multiply to 21, not -21. This option demonstrates the mistake of using the correct numbers but failing to account for the signs properly. This is a classic error where the student didn't consider that the constant term in the original trinomial was negative, indicating that one of the factors must be negative.

Key Takeaways and Tips for Factoring Trinomials

To become a factoring whiz, here are some key takeaways and tips to keep in mind:

  • Identify the pattern: Trinomials of the form x² + bx + c can often be factored into two binomials (x + p)(x + q), where p and q are numbers that multiply to c and add up to b.
  • Find the magic numbers: The most crucial step is to find the two numbers that multiply to the constant term and add up to the coefficient of the linear term.
  • Pay attention to signs: The signs of the numbers are critical. If the constant term is negative, one factor must be positive, and the other must be negative. If the constant term is positive, both factors will have the same sign (either both positive or both negative), which is determined by the sign of the linear term.
  • Double-check your work: Always multiply the factored binomials back together using the FOIL method to ensure you get the original trinomial. This is a simple yet effective way to catch errors.
  • Practice makes perfect: The more you practice factoring, the quicker and more confident you'll become. Start with simple trinomials and gradually move on to more complex ones.

Conclusion: You've Got This!

So, guys, factoring the trinomial x² + 4x - 21 isn't so scary after all! By breaking down the process into smaller steps, understanding the key concepts, and practicing regularly, you can master this important algebraic skill. Remember, the correct answer is D. (x - 3)(x + 7). Keep practicing, and you'll be factoring like a pro in no time! Factoring isn't just about finding the right answer; it's about developing your problem-solving skills and understanding the relationships between algebraic expressions. These skills will be invaluable as you progress in your mathematical journey.

Don't be afraid to make mistakes – they're a natural part of the learning process. Each mistake is an opportunity to learn and grow. So, keep practicing, keep asking questions, and keep exploring the fascinating world of mathematics! You've got this!