How To Factor 56a³ - 17a² - 3a A Step By Step Guide

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Factoring trinomials can seem daunting, but with a systematic approach, even complex expressions like 56a³ - 17a² - 3a can be broken down into simpler components. This guide provides a step-by-step walkthrough of factoring this particular trinomial, along with explanations of the underlying principles and techniques. We'll explore the process in detail, ensuring you understand not just how to factor, but why each step is necessary. Let's dive into the world of factoring and unravel this expression.

1. Identifying Common Factors

When faced with a trinomial like 56a³ - 17a² - 3a, the first and most crucial step is to look for common factors among all the terms. This simplifies the expression and makes subsequent factoring easier. In this case, we can observe that each term contains the variable 'a'. Therefore, 'a' is a common factor. Factoring out 'a' from the trinomial gives us:

a(56a² - 17a - 3)

This initial step is vital because it reduces the complexity of the trinomial we need to factor further. We've effectively transformed a cubic trinomial into a product of a monomial ('a') and a quadratic trinomial (56a² - 17a - 3). Now, our focus shifts to factoring the quadratic expression. Recognizing and extracting common factors is a cornerstone of efficient factoring, preventing unnecessary complications down the line. Remember, always start by looking for the greatest common factor (GCF), which is the largest factor that divides all terms evenly. This ensures that the resulting expression inside the parentheses is in its simplest form, ready for the next stage of factoring.

2. Factoring the Quadratic Trinomial: 56a² - 17a - 3

Now that we've factored out the common factor 'a', we're left with the quadratic trinomial 56a² - 17a - 3. Factoring this quadratic expression requires a different approach, typically involving techniques like the 'ac method' or trial and error. The ac method is particularly useful when the leading coefficient (the coefficient of the a² term) is not 1, as is the case here. Let's delve into the ac method step-by-step:

  1. Identify a, b, and c: In our trinomial, a = 56, b = -17, and c = -3.
  2. Calculate ac: Multiply the leading coefficient (a) by the constant term (c): 56 * -3 = -168.
  3. Find two numbers: We need to find two numbers that multiply to -168 (ac) and add up to -17 (b). This is the crucial step, and it might require some trial and error. The numbers are -24 and 7 because -24 * 7 = -168 and -24 + 7 = -17.
  4. Rewrite the middle term: Replace the middle term (-17a) with the two numbers we found (-24a and 7a): 56a² - 24a + 7a - 3.
  5. Factor by grouping: Now, we have four terms. Group the first two terms and the last two terms together: (56a² - 24a) + (7a - 3).
  6. Factor out the GCF from each group: From the first group, the GCF is 8a. Factoring it out, we get 8a(7a - 3). From the second group, the GCF is 1. Factoring it out, we get 1(7a - 3).
  7. Combine the factors: Notice that both groups now have a common binomial factor (7a - 3). Factor out this common binomial: (7a - 3)(8a + 1).

Therefore, the factored form of the quadratic trinomial 56a² - 17a - 3 is (7a - 3)(8a + 1). This methodical approach ensures that we accurately factor the quadratic expression, setting the stage for the final step of combining all the factors.

3. Combining the Factors for the Complete Factorization

We've successfully factored out the common factor 'a' and then factored the quadratic trinomial 56a² - 17a - 3 into (7a - 3)(8a + 1). Now, to obtain the complete factorization of the original trinomial 56a³ - 17a² - 3a, we need to combine these factors. This involves simply multiplying the common factor 'a' with the factored quadratic expression:

a(56a² - 17a - 3) = a(7a - 3)(8a + 1)

This final expression, a(7a - 3)(8a + 1), represents the completely factored form of the original trinomial. It's essential to understand that this factorization is unique (except for the order of the factors) and represents the simplest form of the expression in terms of its factors. This complete factorization allows us to easily identify the roots or zeros of the polynomial, which are the values of 'a' that make the expression equal to zero. In this case, the roots are a = 0, a = 3/7, and a = -1/8. Combining the individual factors into the complete factorization is the ultimate goal of the factoring process, providing a concise and informative representation of the original expression.

4. Verification and Checking Your Answer

After factoring a trinomial, it's always a good practice to verify your answer. This ensures that you haven't made any errors during the factoring process. The most straightforward way to verify your factored expression is to multiply the factors back together and see if you obtain the original trinomial. In our case, we factored 56a³ - 17a² - 3a into a(7a - 3)(8a + 1). Let's multiply these factors:

  1. Multiply the binomials: First, multiply (7a - 3) and (8a + 1) using the distributive property (also known as the FOIL method): (7a - 3)(8a + 1) = 7a * 8a + 7a * 1 - 3 * 8a - 3 * 1 = 56a² + 7a - 24a - 3 = 56a² - 17a - 3
  2. Multiply by the monomial: Now, multiply the result by 'a': a(56a² - 17a - 3) = 56a³ - 17a² - 3a

Since we obtained the original trinomial, 56a³ - 17a² - 3a, our factorization is correct. Verification is a critical step in the factoring process, as it helps to catch any mistakes and builds confidence in your solution. By systematically multiplying the factors and comparing the result with the original expression, you can ensure the accuracy of your factoring and solidify your understanding of the process. Remember, a little extra time spent verifying your answer can save you from costly errors in subsequent calculations or applications.

5. Conclusion: Mastering Trinomial Factoring

Factoring the trinomial 56a³ - 17a² - 3a involves a series of steps, each building upon the previous one. We started by identifying and factoring out the common factor 'a', which simplified the expression. Then, we tackled the quadratic trinomial 56a² - 17a - 3 using the ac method, a powerful technique for factoring quadratics with a leading coefficient not equal to 1. We found the factors (7a - 3) and (8a + 1) and combined them with the initial common factor to obtain the complete factorization: a(7a - 3)(8a + 1). Finally, we verified our answer by multiplying the factors back together, ensuring that we arrived at the original trinomial.

Mastering trinomial factoring is a fundamental skill in algebra, with applications in solving equations, simplifying expressions, and understanding polynomial functions. By following a systematic approach, such as the one outlined in this guide, you can confidently factor even complex trinomials. Remember to always look for common factors first, choose the appropriate factoring technique for the resulting expression, and verify your answer to ensure accuracy. With practice and a solid understanding of the underlying principles, you'll be well-equipped to tackle any factoring challenge that comes your way. The ability to factor trinomials efficiently opens doors to more advanced algebraic concepts and problem-solving strategies, making it a valuable asset in your mathematical toolkit.