FOIL Method Polynomial Multiplication Solving (x+5)(x²-3x)

#Understanding the FOIL Method

In the realm of algebra, multiplying polynomials is a fundamental skill. Among the various techniques available, the FOIL method stands out as a straightforward and effective approach, especially when dealing with the product of two binomials. This method, an acronym for First, Outer, Inner, Last, provides a structured way to ensure that each term in the first binomial is multiplied by each term in the second binomial. By systematically applying the FOIL method, you can confidently expand polynomial expressions and simplify them into a standard form. In this comprehensive guide, we will delve into the intricacies of the FOIL method, demonstrating its application through a detailed walkthrough of the problem (x+5)(x²-3x). This example will not only illustrate the practical steps involved but also reinforce the underlying principles, empowering you to tackle similar problems with ease and accuracy. Remember, mastering the FOIL method is not just about memorizing an acronym; it's about developing a deep understanding of polynomial multiplication, a skill that will serve you well in more advanced mathematical concepts. So, let's embark on this journey together, unraveling the power and elegance of the FOIL method.

Before we jump into the application of the FOIL method, let's first dissect the problem at hand: (x+5)(x²-3x). This expression represents the product of two polynomial expressions: a binomial (x+5) and another binomial-like expression (x²-3x). Although the second expression might not immediately appear as a standard binomial, it's crucial to recognize that the FOIL method can be adapted to handle such cases. The key is to treat (x²-3x) as a single entity with two terms, x² and -3x. Understanding the structure of the expressions we're dealing with is the first step towards successfully applying the FOIL method. Now, let's break down each term in the binomial (x+5) and the expression (x²-3x). In the binomial (x+5), we have two terms: 'x' and '+5'. Similarly, in the expression (x²-3x), we have two terms: 'x²' and '-3x'. These individual terms will be the building blocks for our multiplication process. By carefully identifying each term, we set the stage for a systematic application of the FOIL method. In the following sections, we will meticulously walk through each step of the FOIL process, ensuring that every term is correctly multiplied and combined. This methodical approach will not only lead us to the correct solution but also instill a sense of confidence in our algebraic manipulations.

Now, let's dive into the heart of the FOIL method and apply it to our problem: (x+5)(x²-3x). Remember, FOIL stands for First, Outer, Inner, Last, which represents the order in which we will multiply the terms. Let's break it down step by step:

1. First: Multiply the first terms of each binomial. In our case, the first terms are 'x' from (x+5) and 'x²' from (x²-3x). Multiplying these terms gives us: x * x² = x³. This is the first piece of our puzzle, and it sets the stage for the subsequent steps.

2. Outer: Multiply the outer terms of the expression. The outer terms are 'x' from (x+5) and '-3x' from (x²-3x). Multiplying these gives us: x * -3x = -3x². It's crucial to pay attention to the signs here, as the negative sign will affect the final result.

3. Inner: Multiply the inner terms of the expression. The inner terms are '+5' from (x+5) and 'x²' from (x²-3x). Multiplying these gives us: 5 * x² = 5x². This step is straightforward, and we're steadily building towards the complete expansion.

4. Last: Multiply the last terms of each binomial. The last terms are '+5' from (x+5) and '-3x' from (x²-3x). Multiplying these gives us: 5 * -3x = -15x. Again, the negative sign plays a crucial role in determining the correct result.

By systematically applying the FOIL method, we have successfully multiplied each term in the first expression by each term in the second expression. Now, we have the expanded form: x³ - 3x² + 5x² - 15x. The next step is to simplify this expression by combining like terms, which we will tackle in the following section.

After applying the FOIL method to (x+5)(x²-3x), we arrived at the expanded form: x³ - 3x² + 5x² - 15x. Now, the next crucial step is to simplify this expression by combining like terms. Like terms are those that have the same variable raised to the same power. In our expanded form, we can identify two terms that fit this criterion: -3x² and +5x². These terms both have the variable 'x' raised to the power of 2, making them like terms. To combine these like terms, we simply add their coefficients. The coefficient is the numerical factor that multiplies the variable. In this case, the coefficients are -3 and +5. Adding these coefficients gives us: -3 + 5 = 2. Therefore, when we combine -3x² and +5x², we get 2x². Now, let's rewrite our expression, replacing the like terms with their combined form: x³ + 2x² - 15x. Notice that the terms x³ and -15x do not have any like terms in the expression. This means they remain unchanged in the simplified form. The process of combining like terms is a fundamental step in simplifying polynomial expressions. It allows us to reduce the expression to its most concise form, making it easier to analyze and work with. By carefully identifying and combining like terms, we ensure that our final answer is accurate and represents the simplified form of the original expression. In the next section, we will present our final answer and discuss its significance in the context of the original problem.

After meticulously applying the FOIL method and combining like terms, we have arrived at the final simplified product of (x+5)(x²-3x): x³ + 2x² - 15x. This result represents the expanded form of the original expression, where all possible multiplications have been performed and like terms have been combined. Let's take a moment to appreciate the journey we've undertaken. We started with a seemingly complex expression involving the product of two polynomials. By systematically applying the FOIL method, we broke down the problem into manageable steps. We multiplied each term in the first expression by each term in the second expression, carefully tracking the signs and exponents. Then, we identified and combined like terms, simplifying the expression to its most concise form. The final product, x³ + 2x² - 15x, is a polynomial expression with three terms: x³, 2x², and -15x. Each term has a specific coefficient and a variable raised to a certain power. This final answer is not just a numerical result; it's a symbolic representation of the relationship between the original expressions. It encapsulates the complete multiplication process and provides a clear and unambiguous representation of the product. In the context of the original question, this is the definitive answer. It demonstrates our mastery of the FOIL method and our ability to simplify polynomial expressions. In the concluding section, we will briefly summarize the key takeaways from this problem and reinforce the importance of the FOIL method in algebraic manipulations.

In this comprehensive guide, we have successfully navigated the process of multiplying polynomials using the FOIL method. We tackled the problem (x+5)(x²-3x), demonstrating a step-by-step approach that involved expanding the expression and simplifying it by combining like terms. The FOIL method, an acronym for First, Outer, Inner, Last, proved to be a powerful tool for systematically multiplying binomials and similar expressions. By following this method, we ensured that every term in the first expression was multiplied by every term in the second expression, leaving no room for error. We also emphasized the importance of careful attention to signs and exponents, as these details can significantly impact the final result. Furthermore, we highlighted the significance of combining like terms, a crucial step in simplifying polynomial expressions and presenting them in their most concise form. The final answer, x³ + 2x² - 15x, represents the fully expanded and simplified product of the original expression. This result is not just a numerical value; it's a symbolic representation of the relationship between the polynomials. Mastering the FOIL method is a fundamental skill in algebra. It provides a structured approach to polynomial multiplication, enabling you to tackle complex expressions with confidence. By understanding the underlying principles and practicing the steps involved, you can develop a strong foundation for more advanced mathematical concepts. In conclusion, the FOIL method is an invaluable tool in your algebraic arsenal. It empowers you to expand and simplify polynomial expressions with ease and accuracy. By mastering this method, you not only solve problems effectively but also gain a deeper understanding of the fundamental principles of algebra.

Therefore, the correct answer is D. x³ + 2x² - 15x