Factoring Completely 50a²b⁵ - 35a⁴b³ + 5a³b⁴ A Step-by-Step Guide

Factoring polynomials is a fundamental skill in algebra, allowing us to simplify expressions, solve equations, and gain deeper insights into mathematical relationships. In this comprehensive guide, we will delve into the process of factoring the polynomial 50a²b⁵ - 35a⁴b³ + 5a³b⁴ completely. We will break down each step, providing clear explanations and examples to ensure a thorough understanding. This article aims to equip you with the knowledge and confidence to tackle similar factoring problems with ease. We'll explore the initial steps of identifying the greatest common factor (GCF), then proceed to factor it out, and finally examine the resulting expression for further factorization opportunities. By the end of this guide, you'll not only understand how to factor this specific polynomial but also grasp the underlying principles applicable to a wide range of factoring scenarios. Whether you're a student learning algebra or someone looking to refresh your math skills, this guide offers a step-by-step approach to mastering polynomial factorization. So, let's embark on this journey of mathematical exploration and unlock the secrets of factoring polynomials completely. Throughout this guide, we'll emphasize the importance of checking your work and ensuring that the factored form is indeed equivalent to the original polynomial. This crucial step helps to minimize errors and solidify your understanding of the factoring process. By focusing on both the mechanics of factoring and the underlying concepts, this guide aims to provide a holistic learning experience. Let's dive in and begin the process of factoring the polynomial 50a²b⁵ - 35a⁴b³ + 5a³b⁴ completely.

Identifying the Greatest Common Factor (GCF)

To begin factoring the polynomial 50a²b⁵ - 35a⁴b³ + 5a³b⁴, our first crucial step is identifying the greatest common factor (GCF) of the terms. The GCF is the largest factor that divides evenly into all the terms of the polynomial. Finding the GCF involves examining both the coefficients (the numerical parts) and the variables (the literal parts) of each term. Let's break down the process step by step. First, we consider the coefficients: 50, -35, and 5. We need to find the largest number that divides evenly into all three. The factors of 50 are 1, 2, 5, 10, 25, and 50. The factors of 35 are 1, 5, 7, and 35. The factors of 5 are 1 and 5. Comparing these factors, we see that the greatest common factor of the coefficients is 5. Next, we turn our attention to the variables. We have the variables a and b, each raised to different powers in the three terms. For the variable 'a', we have a², a⁴, and a³. The GCF for variables is the variable raised to the lowest power present in any of the terms. In this case, the lowest power of 'a' is a², so a² is part of our GCF. For the variable 'b', we have b⁵, b³, and b⁴. Similarly, we look for the lowest power of 'b', which is b³. Thus, b³ is also part of our GCF. Now, we combine the GCF of the coefficients and the GCF of the variables to find the overall GCF of the polynomial. The GCF is 5a²b³. This means that 5a²b³ is the largest expression that divides evenly into each term of the polynomial 50a²b⁵ - 35a⁴b³ + 5a³b⁴. Identifying the GCF is a fundamental step in factoring because it allows us to simplify the polynomial by factoring it out. Once we have the GCF, we can proceed to the next step of dividing each term of the polynomial by the GCF. This process will reveal the remaining factors and bring us closer to completely factoring the polynomial. Understanding how to find the GCF is essential for mastering polynomial factorization. It's a skill that will be used repeatedly in more advanced algebraic manipulations. So, let's move on to the next step and see how we can use this GCF to factor the polynomial further.

Factoring Out the GCF

Having identified the greatest common factor (GCF) as 5a²b³, the next crucial step in factoring the polynomial 50a²b⁵ - 35a⁴b³ + 5a³b⁴ is to factor it out. Factoring out the GCF involves dividing each term of the polynomial by the GCF and writing the polynomial as a product of the GCF and the resulting expression. This process effectively reverses the distributive property, allowing us to simplify the polynomial and make it easier to work with. To begin, we divide each term of the polynomial by 5a²b³: (50a²b⁵) / (5a²b³) = 10b² (-35a⁴b³) / (5a²b³) = -7a² (5a³b⁴) / (5a²b³) = ab Now, we can rewrite the original polynomial as the product of the GCF and the expression formed by the quotients we just calculated. This gives us: 5a²b³(10b² - 7a² + ab) This expression is equivalent to the original polynomial, but it is now in a factored form. The GCF, 5a²b³, is factored out, and the remaining expression, (10b² - 7a² + ab), is enclosed in parentheses. Factoring out the GCF is a significant step in simplifying the polynomial. It reduces the complexity of the expression inside the parentheses, which may make it easier to identify further factoring opportunities. For instance, the expression (10b² - 7a² + ab) might be factorable itself, depending on its structure. However, it's important to note that factoring out the GCF is always the first step in completely factoring a polynomial. It ensures that we have extracted the largest common factor, leaving us with the simplest possible expression to work with. Moreover, factoring out the GCF can also help in solving polynomial equations. By setting the factored polynomial equal to zero, we can use the zero-product property to find the roots of the equation. In this case, factoring out 5a²b³ allows us to immediately identify that a = 0 and b = 0 are solutions to the equation 50a²b⁵ - 35a⁴b³ + 5a³b⁴ = 0. Factoring out the GCF is a fundamental technique in algebra, and mastering it is crucial for success in more advanced mathematical concepts. So, with the GCF factored out, let's move on to the next step and investigate whether the remaining expression, (10b² - 7a² + ab), can be factored further.

Examining the Resulting Expression for Further Factorization

After factoring out the greatest common factor (GCF) from the polynomial 50a²b⁵ - 35a⁴b³ + 5a³b⁴, we arrived at the expression 5a²b³(10b² - 7a² + ab). Now, the crucial next step is to examine the expression inside the parentheses, (10b² - 7a² + ab), to determine if it can be factored further. This is a critical part of completely factoring a polynomial, as it ensures that we have broken it down into its simplest possible factors. The expression (10b² - 7a² + ab) is a trinomial, a polynomial with three terms. To determine if it can be factored, we need to consider various factoring techniques. One common method for factoring trinomials is to look for patterns such as the difference of squares, perfect square trinomials, or simple quadratic trinomials. However, in this case, the trinomial (10b² - 7a² + ab) does not fit any of these standard patterns directly. The presence of both a² and b² terms, along with the mixed term 'ab', suggests that we might need to explore other factoring techniques, such as grouping or trial and error. Grouping involves rearranging the terms and looking for common factors within groups of terms. However, in this case, grouping does not lead to any obvious factorization. Trial and error involves attempting different combinations of factors to see if they multiply to give the original trinomial. This can be a more time-consuming process, but it is sometimes necessary when other methods fail. In this particular case, after careful consideration and exploration of different factoring techniques, it becomes apparent that the trinomial (10b² - 7a² + ab) cannot be factored further using elementary methods. This does not mean that the expression is prime (unfactorable), but it does mean that we have exhausted the common factoring techniques typically taught in introductory algebra courses. In more advanced contexts, there might be more sophisticated techniques or methods to attempt factorization, but within the scope of basic algebra, the trinomial (10b² - 7a² + ab) is considered to be unfactorable. Therefore, the completely factored form of the original polynomial 50a²b⁵ - 35a⁴b³ + 5a³b⁴ is 5a²b³(10b² - 7a² + ab). This is the final answer, as we have factored out the GCF and determined that the remaining expression cannot be factored further. Checking your work is an essential step to ensure the accuracy of your factoring. You can verify that the factored form is equivalent to the original polynomial by distributing the GCF back into the parentheses. If the result matches the original polynomial, then you have factored it correctly. In this case, multiplying 5a²b³ by (10b² - 7a² + ab) gives us back 50a²b⁵ - 35a⁴b³ + 5a³b⁴, confirming that our factoring is correct.

Final Factored Form and Verification

After carefully following the steps of identifying the greatest common factor (GCF), factoring it out, and examining the resulting expression for further factorization, we have arrived at the completely factored form of the polynomial 50a²b⁵ - 35a⁴b³ + 5a³b⁴. Our journey began with the original polynomial and led us through a systematic process of simplification and factorization. We first identified the GCF as 5a²b³, which is the largest expression that divides evenly into all the terms of the polynomial. This step is crucial because it allows us to reduce the complexity of the polynomial and make it easier to work with. Next, we factored out the GCF, dividing each term of the polynomial by 5a²b³ and rewriting the polynomial as a product of the GCF and the resulting expression. This gave us 5a²b³(10b² - 7a² + ab). This expression is equivalent to the original polynomial, but it is now in a partially factored form. The GCF is factored out, and the remaining expression, (10b² - 7a² + ab), is enclosed in parentheses. Then, we turned our attention to the expression inside the parentheses, (10b² - 7a² + ab), to determine if it could be factored further. We explored various factoring techniques, such as looking for patterns like the difference of squares, perfect square trinomials, or simple quadratic trinomials. We also considered grouping and trial and error methods. However, after careful consideration and exploration, we concluded that the trinomial (10b² - 7a² + ab) cannot be factored further using elementary methods. This means that we have reached the final step in the factorization process. The polynomial is now completely factored, and we have broken it down into its simplest possible factors. The completely factored form of the polynomial 50a²b⁵ - 35a⁴b³ + 5a³b⁴ is 5a²b³(10b² - 7a² + ab). This is our final answer. However, to ensure that our factoring is correct, it is essential to verify our result. We can do this by distributing the GCF back into the parentheses and checking if the result matches the original polynomial. Multiplying 5a²b³ by (10b² - 7a² + ab) gives us: 5a²b³ * 10b² = 50a²b⁵ 5a²b³ * (-7a²) = -35a⁴b³ 5a²b³ * ab = 5a³b⁴ Adding these terms together, we get: 50a²b⁵ - 35a⁴b³ + 5a³b⁴ This is exactly the original polynomial, which confirms that our factoring is correct. Therefore, we can confidently state that the completely factored form of 50a²b⁵ - 35a⁴b³ + 5a³b⁴ is indeed 5a²b³(10b² - 7a² + ab). This comprehensive process demonstrates the importance of systematically approaching factoring problems, starting with identifying the GCF and progressing through the necessary steps to ensure complete factorization. Factoring is a fundamental skill in algebra, and mastering it is crucial for success in more advanced mathematical concepts.