Factoring The Trinomial -7x² - 5x + 18 A Step-by-Step Guide

by ADMIN 60 views

Factoring trinomials can sometimes feel like solving a puzzle, but with the right strategies, it becomes a manageable task. In this comprehensive guide, we'll break down the process of factoring the trinomial -7x² - 5x + 18. We'll explore different approaches, analyze potential solutions, and provide a step-by-step explanation to arrive at the correct factorization. Whether you're a student grappling with algebra or simply looking to refresh your factoring skills, this article will equip you with the knowledge and confidence to tackle similar problems.

Understanding the Problem

The trinomial we need to factor is -7x² - 5x + 18. Factoring a trinomial involves expressing it as a product of two binomials. In other words, we want to find two expressions of the form (ax + b) and (cx + d) such that when multiplied together, they equal the original trinomial. This process is the reverse of expanding binomials, and it's a fundamental skill in algebra.

Key Concepts: Before we dive into the solution, let's quickly review some key concepts. A trinomial is a polynomial with three terms. Factoring is the process of decomposing a polynomial into a product of simpler polynomials. The distributive property (or FOIL method) is used to expand binomials, and we'll use it in reverse to factor trinomials.

Methods for Factoring Trinomials

There are several methods for factoring trinomials, and the best approach often depends on the specific trinomial. Some common methods include:

  • Trial and Error: This method involves making educated guesses about the binomial factors and then checking if their product matches the original trinomial. It can be time-consuming but is effective with practice.
  • The AC Method: This method is more systematic and involves finding two numbers that multiply to the product of the leading coefficient (A) and the constant term (C) and add up to the middle coefficient (B). It's particularly useful for trinomials with a leading coefficient other than 1.
  • Factoring by Grouping: This method is often used in conjunction with the AC method. After finding the two numbers, we rewrite the middle term of the trinomial as a sum of two terms and then factor by grouping.

We'll primarily use the AC method and factoring by grouping to solve this problem, but we'll also consider a trial-and-error approach to verify our solution.

Step-by-Step Solution

Let's apply the AC method to factor the trinomial -7x² - 5x + 18. Here's a detailed breakdown:

1. Identify A, B, and C: In the trinomial -7x² - 5x + 18:

  • A = -7 (the coefficient of x²)
  • B = -5 (the coefficient of x)
  • C = 18 (the constant term)

2. Calculate AC: Multiply A and C: AC = (-7) * (18) = -126

3. Find Two Numbers: We need to find two numbers that multiply to -126 and add up to B (-5). This is the crucial step, and it may require some thought and experimentation. Let's list some factor pairs of -126:

  • 1 and -126
  • -1 and 126
  • 2 and -63
  • -2 and 63
  • 3 and -42
  • -3 and 42
  • 6 and -21
  • -6 and 21
  • 7 and -18
  • -7 and 18
  • 9 and -14
  • -9 and 14

Among these pairs, -14 and 9 satisfy our conditions: (-14) * (9) = -126 and (-14) + (9) = -5.

4. Rewrite the Middle Term: Now, we rewrite the middle term (-5x) using the two numbers we found (-14 and 9):

-7x² - 5x + 18 = -7x² - 14x + 9x + 18

5. Factor by Grouping: Group the terms in pairs and factor out the greatest common factor (GCF) from each pair:

  • From the first pair, -7x² - 14x, the GCF is -7x. Factoring this out gives: -7x(x + 2)
  • From the second pair, 9x + 18, the GCF is 9. Factoring this out gives: 9(x + 2)

So, our expression becomes: -7x(x + 2) + 9(x + 2)

6. Factor out the Common Binomial: Notice that both terms now have a common binomial factor of (x + 2). Factor this out:

(x + 2)(-7x + 9)

7. Final Factorization: Therefore, the factored form of -7x² - 5x + 18 is (x + 2)(-7x + 9).

Analyzing the Answer Choices

Now that we have the factored form, let's compare it to the given answer choices:

A. -7(x - 6)(x + 1) B. -1(7x - 9)(x + 2) C. (-7x + 9)(x - 2) D. (-7x - 9)(x + 2)

Our factorization is (x + 2)(-7x + 9), which is the same as (-7x + 9)(x + 2) due to the commutative property of multiplication. Therefore, answer choice B. -1(7x - 9)(x + 2) is the correct factorization. To see this, distribute the -1 into the first binomial: -1(7x - 9) = -7x + 9. So, -1(7x - 9)(x + 2) = (-7x + 9)(x + 2).

Answer choice C is close, but the sign is incorrect in the second binomial. Let's explore why the other options are incorrect.

  • Option A: -7(x - 6)(x + 1) Expanding this gives: -7(x² + x - 6x - 6) = -7(x² - 5x - 6) = -7x² + 35x + 42, which is not the original trinomial.

  • Option D: (-7x - 9)(x + 2) Expanding this gives: -7x² - 14x - 9x - 18 = -7x² - 23x - 18, which is also not the original trinomial.

Common Mistakes to Avoid

Factoring trinomials can be tricky, and it's easy to make mistakes. Here are some common errors to watch out for:

  • Sign Errors: Pay close attention to the signs of the coefficients and constants. A simple sign error can lead to an incorrect factorization.
  • Incorrectly Identifying Factors: Make sure the two numbers you find multiply to AC and add up to B. Double-check your calculations.
  • Forgetting to Factor Completely: Sometimes, after factoring by grouping, there may be a common factor that can be factored out further. Ensure you've factored the trinomial completely.
  • Rushing the Process: Factoring takes time and careful attention to detail. Avoid rushing, and double-check each step.

Tips and Tricks for Factoring Trinomials

Here are some helpful tips and tricks to improve your factoring skills:

  • Practice Regularly: The more you practice, the better you'll become at recognizing patterns and applying factoring techniques.
  • Use the AC Method: The AC method is a reliable approach for factoring trinomials, especially when the leading coefficient is not 1.
  • Check Your Work: After factoring, multiply the binomials to verify that their product matches the original trinomial.
  • Look for GCF First: Before attempting any other factoring method, check if there is a greatest common factor that can be factored out from all terms. This simplifies the trinomial and makes it easier to factor.
  • Recognize Special Cases: Be on the lookout for special cases like perfect square trinomials and differences of squares. These have specific factoring patterns that can save time.

Conclusion

Factoring the trinomial -7x² - 5x + 18 requires a systematic approach, careful attention to detail, and a solid understanding of factoring techniques. By using the AC method and factoring by grouping, we arrived at the correct factorization: (-7x + 9)(x + 2). It's crucial to compare the final factorization with the given answer choices and verify the solution by expanding the binomials. Remember to practice regularly, avoid common mistakes, and utilize helpful tips and tricks to enhance your factoring skills. With dedication and the right strategies, you'll master the art of factoring trinomials.

This article has provided a comprehensive guide to factoring the trinomial -7x² - 5x + 18. By understanding the underlying concepts, following the step-by-step solution, and avoiding common mistakes, you can confidently tackle similar problems. Happy factoring!