Factoring Trinomials Completely A Step-by-Step Guide

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Factoring trinomials can be a daunting task, especially when dealing with a leading coefficient other than 1. However, with a systematic approach, you can master the art of factoring and simplify complex expressions. In this article, we will delve into factoring the trinomial -6x² + 26x + 20 completely. We will break down each step, providing clear explanations and examples to guide you through the process. Understanding the fundamentals of factoring is crucial not only for algebraic manipulations but also for solving quadratic equations and tackling various mathematical problems.

1. Identifying the Trinomial and the Greatest Common Factor (GCF)

Before diving into the factoring process, it's essential to correctly identify the trinomial. A trinomial is a polynomial expression consisting of three terms. In our case, the trinomial is -6x² + 26x + 20. Now, the first step in factoring any polynomial is to look for the greatest common factor (GCF) of all the terms. The GCF is the largest factor that divides all the terms without leaving a remainder. For the trinomial -6x² + 26x + 20, the coefficients are -6, 26, and 20. The GCF of these numbers is 2. Also, notice that all the coefficients are even, and the leading coefficient is negative. Factoring out a -2 will make the subsequent steps easier. Therefore, we factor out -2 from each term:

-6x² + 26x + 20 = -2(3x² - 13x - 10)

This simplifies the trinomial and makes it easier to work with. Now, we focus on factoring the quadratic expression inside the parentheses, which is 3x² - 13x - 10. This step is vital because it reduces the complexity of the trinomial and prepares it for further factoring. By identifying and extracting the GCF, we've laid a solid foundation for the next steps in our factoring journey. Remember, this initial step can significantly simplify the process and prevent errors later on.

2. Factoring the Simplified Trinomial: 3x² - 13x - 10

Now that we've factored out the GCF, we're left with the simplified trinomial 3x² - 13x - 10. Factoring this trinomial involves finding two binomials that, when multiplied together, give us the original trinomial. One common method for factoring trinomials of the form ax² + bx + c is the AC method. In this method, we multiply the leading coefficient (a) by the constant term (c), which in our case is 3 * (-10) = -30. Next, we need to find two numbers that multiply to -30 and add up to the middle coefficient (b), which is -13.

The pairs of factors of -30 are: (1, -30), (-1, 30), (2, -15), (-2, 15), (3, -10), (-3, 10), (5, -6), and (-5, 6). Among these pairs, the numbers 2 and -15 satisfy our conditions: 2 * (-15) = -30 and 2 + (-15) = -13. Now, we rewrite the middle term (-13x) using these two numbers:

3x² - 13x - 10 = 3x² + 2x - 15x - 10

Next, we factor by grouping. We group the first two terms and the last two terms:

(3x² + 2x) + (-15x - 10)

Factor out the GCF from each group:

x(3x + 2) - 5(3x + 2)

Notice that both terms now have a common binomial factor, which is (3x + 2). Factor out this common binomial:

(3x + 2)(x - 5)

So, the factored form of 3x² - 13x - 10 is (3x + 2)(x - 5). This step is crucial as it breaks down the quadratic expression into a product of two binomials, making it much easier to solve and analyze. Factoring by grouping allows us to systematically break down the trinomial and identify the binomial factors.

3. Combining the GCF and Factored Binomials

Now that we have factored the simplified trinomial 3x² - 13x - 10 as (3x + 2)(x - 5), we need to remember the GCF that we factored out in the first step. The GCF was -2. To completely factor the original trinomial -6x² + 26x + 20, we need to include the GCF in our final factored form. Therefore, we multiply the GCF (-2) by the factored binomials:

-2(3x + 2)(x - 5)

This is the completely factored form of the original trinomial. It's crucial to include the GCF in the final answer, as it represents an essential part of the original expression. Without the GCF, the factored form would not be equivalent to the original trinomial. Always double-check your work by distributing the factored form to ensure it matches the original trinomial.

4. Verification: Expanding the Factored Form

To ensure that our factoring is correct, we can expand the factored form and check if it matches the original trinomial. This step is a crucial part of the factoring process, as it helps prevent errors and ensures the accuracy of our solution. We start with the factored form:

-2(3x + 2)(x - 5)

First, we multiply the two binomials (3x + 2) and (x - 5) using the distributive property (also known as the FOIL method):

(3x + 2)(x - 5) = 3x * x + 3x * (-5) + 2 * x + 2 * (-5)

= 3x² - 15x + 2x - 10

= 3x² - 13x - 10

Now, we multiply the result by the GCF, which is -2:

-2(3x² - 13x - 10) = -2 * 3x² + (-2) * (-13x) + (-2) * (-10)

= -6x² + 26x + 20

This matches our original trinomial, -6x² + 26x + 20. Therefore, our factoring is correct. Verification is an essential step in the factoring process. By expanding the factored form and comparing it to the original trinomial, we can be confident in our solution and avoid costly mistakes. This step solidifies our understanding and ensures the accuracy of our work.

5. Analyzing the Answer Choices and Selecting the Correct Option

Now that we have completely factored the trinomial and verified our result, we can confidently analyze the given answer choices and select the correct option. Our factored form is:

-2(3x + 2)(x - 5)

Looking at the answer choices provided:

A. -2(3x + 5)(x + 2) B. -2(3x² - 13x + 10) C. -2(3x² - 13x - 10) D. -2(3x - 10)(x + 1)

None of the provided answer choices exactly match our factored form, -2(3x + 2)(x - 5). However, if we take a closer look at our factored form, we realize that one of the binomials has a sign difference. Specifically, in choice C, if we factor the quadratic expression, we should get:

−6x2+26x+20 −2(3x2−13x−10) −2(3x+2)(x−5)-6 x^2+26 x+20 \ -2(3 x^2-13 x-10) \ -2(3 x+2)(x-5)

Which means that None of the options are correct. It is important to carefully analyze each option and compare it with our result.

6. Common Mistakes to Avoid When Factoring Trinomials

Factoring trinomials can be tricky, and it's easy to make mistakes if you're not careful. Here are some common mistakes to avoid:

  1. Forgetting to factor out the GCF: Always look for the greatest common factor (GCF) first. Factoring out the GCF simplifies the trinomial and makes it easier to factor. Forgetting this step can lead to more complex factoring or incorrect answers.
  2. Incorrectly identifying factor pairs: When using the AC method, make sure you find the correct pair of factors that multiply to the product of a and c and add up to b. An incorrect pair will result in incorrect binomial factors.
  3. Sign errors: Pay close attention to the signs of the terms. A small sign error can completely change the factored form. Always double-check your signs during each step of the factoring process.
  4. Incorrectly factoring by grouping: When factoring by grouping, ensure that the binomial factors are the same in each group. If they are not, you may have made an error in a previous step.
  5. Not verifying the answer: Always verify your factored form by expanding it and comparing it to the original trinomial. This step helps catch any errors and ensures the accuracy of your solution.
  6. Rushing through the process: Factoring requires careful attention to detail. Avoid rushing, and take your time to ensure each step is correct.

By being aware of these common mistakes and taking steps to avoid them, you can improve your factoring skills and solve problems more accurately. Practice is key to mastering factoring, so work through plenty of examples and learn from any mistakes you make.

7. Practice Problems and Further Learning Resources

To master factoring trinomials, consistent practice is essential. Working through a variety of problems will help you solidify your understanding of the concepts and improve your problem-solving skills. Here are some practice problems you can try:

  1. Factor 2x² + 7x + 3 completely.
  2. Factor -4x² + 8x + 5 completely.
  3. Factor 6x² - 11x - 10 completely.
  4. Factor 9x² - 24x + 16 completely.
  5. Factor -12x² + 28x - 15 completely.

Additionally, there are many further learning resources available online and in textbooks that can help you deepen your understanding of factoring. Websites like Khan Academy, YouTube channels dedicated to mathematics, and online math forums can provide valuable explanations, examples, and practice problems. Textbooks and workbooks often include detailed lessons and practice exercises as well.

By combining practice with these resources, you can build a strong foundation in factoring trinomials and tackle more complex algebraic problems with confidence. Don't be afraid to seek out additional help and explanations as needed. Consistent effort and a willingness to learn will lead to success in factoring and beyond.

Conclusion

Factoring trinomials completely is a fundamental skill in algebra that opens doors to solving a wide range of mathematical problems. In this article, we've walked through a step-by-step process for factoring the trinomial -6x² + 26x + 20, emphasizing the importance of identifying the GCF, using the AC method, factoring by grouping, and verifying the answer. We've also highlighted common mistakes to avoid and provided practice problems and resources for further learning. Mastering factoring not only enhances your algebraic abilities but also strengthens your problem-solving skills in general. By consistently practicing and applying these techniques, you'll become more confident and proficient in factoring trinomials and tackling more advanced mathematical concepts.