Multiplying Binomials: Step-by-Step Guide (x+3)(x-9)

by Alex Johnson 53 views

Welcome to this comprehensive guide on multiplying binomials! In this article, we will delve into the process of multiplying two binomials, specifically focusing on the example (x+3)(x-9). Mastering this skill is crucial for anyone studying algebra and higher-level mathematics. We'll break down the steps in an easy-to-understand manner, ensuring you grasp the underlying principles. So, let's dive in and explore the world of binomial multiplication!

Understanding Binomials

Before we jump into the multiplication process, it's important to understand what binomials are. In mathematics, a binomial is a polynomial expression with exactly two terms. These terms are often connected by an addition or subtraction sign. For instance, in the expression (x+3) and (x-9), we have two binomials. The terms 'x' and '3' (or 'x' and '-9') are the individual components of each binomial. Understanding this fundamental concept is key to successfully multiplying these expressions. We need to recognize the structure of binomials to apply the correct methods for their multiplication, which we will discuss in detail in the following sections.

The Significance of Binomial Multiplication in Mathematics

Binomial multiplication is not just a standalone mathematical operation; it's a building block for more complex algebraic concepts. It forms the basis for polynomial factorization, solving quadratic equations, and understanding various mathematical models. When you grasp how to multiply binomials, you're essentially unlocking a door to a broader understanding of algebraic manipulations. This skill is particularly crucial in fields like engineering, physics, and computer science, where mathematical models often involve polynomial expressions. Therefore, mastering binomial multiplication is an investment in your mathematical journey, paving the way for tackling more advanced problems and concepts with confidence and precision.

The FOIL Method: A Key Technique

One of the most common and effective methods for multiplying binomials is the FOIL method. FOIL is an acronym that stands for First, Outer, Inner, Last. It provides a systematic approach to ensure that each term in the first binomial is multiplied by each term in the second binomial. This method is particularly useful for beginners as it breaks down the multiplication process into manageable steps. By following the FOIL method, you can avoid missing any term multiplications and maintain the correct order of operations. This structured approach not only simplifies the multiplication process but also reduces the chances of making errors, leading to more accurate and reliable results in your algebraic calculations.

Breaking Down the FOIL Steps

Let's break down each step of the FOIL method with our example (x+3)(x-9):

  1. First: Multiply the first terms in each binomial. In our example, this means multiplying 'x' from the first binomial by 'x' from the second binomial: x * x = x². This step sets the foundation for the resulting polynomial expression, capturing the highest degree term.
  2. Outer: Multiply the outer terms in the expression. This involves multiplying 'x' from the first binomial by '-9' from the second binomial: x * -9 = -9x. This step introduces a linear term, which contributes to the overall behavior of the polynomial.
  3. Inner: Multiply the inner terms. This means multiplying '3' from the first binomial by 'x' from the second binomial: 3 * x = 3x. Similar to the outer terms, this step also contributes a linear term, which will later be combined with the outer term to simplify the expression.
  4. Last: Multiply the last terms in each binomial. Here, we multiply '3' from the first binomial by '-9' from the second binomial: 3 * -9 = -27. This final step introduces the constant term, which is crucial for defining the polynomial's position and value on the coordinate plane.

By meticulously following these steps, the FOIL method ensures that no term is left unmultiplied, leading to a complete and accurate expansion of the binomial expression. The next step involves combining these individual results to form the final product.

Applying the FOIL Method to (x+3)(x-9)

Now that we've understood the FOIL method, let's apply it step-by-step to our example, (x+3)(x-9). This practical application will solidify your understanding of the method and demonstrate how it translates into a tangible result.

  1. First: Multiply the first terms: x * x = x²
  2. Outer: Multiply the outer terms: x * -9 = -9x
  3. Inner: Multiply the inner terms: 3 * x = 3x
  4. Last: Multiply the last terms: 3 * -9 = -27

By following the FOIL method, we have expanded the binomials into individual terms. Now, we have the expression: x² - 9x + 3x - 27. This is a crucial intermediate step that lays the groundwork for simplifying the expression further. The next step involves combining like terms to arrive at the final simplified polynomial.

Combining Like Terms for Simplification

After applying the FOIL method, we have the expression x² - 9x + 3x - 27. The next crucial step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression, -9x and 3x are like terms because they both contain the variable 'x' raised to the power of 1. Combining like terms simplifies the expression, making it easier to work with and understand. This process involves adding or subtracting the coefficients (the numerical part of the term) of the like terms while keeping the variable part the same. This simplification is not just a matter of mathematical neatness; it's essential for solving equations, graphing functions, and further algebraic manipulations. By reducing the expression to its simplest form, we can more easily analyze its properties and use it in various mathematical contexts.

Step-by-Step Combination

To combine the like terms in our expression, we focus on the terms -9x and +3x. We add their coefficients: -9 + 3 = -6. Therefore, -9x + 3x simplifies to -6x. Now, we rewrite the entire expression with this simplification. The x² term and the constant term -27 remain unchanged as there are no other like terms to combine with them. This step is crucial for presenting the polynomial in its most concise and understandable form. The result of this combination is a simplified quadratic expression, which reveals the polynomial's structure and behavior more clearly.

The Final Result

After combining like terms, our expression x² - 9x + 3x - 27 simplifies to x² - 6x - 27. This is the final result of multiplying the binomials (x+3)(x-9). This quadratic expression represents the product of the two original binomials. It's important to note that this simplified form is not only more compact but also more useful for various mathematical applications. For instance, we can now easily analyze the roots of this quadratic equation, graph the corresponding parabola, or use it in further algebraic calculations. Understanding how to arrive at this final result is a testament to your grasp of binomial multiplication and the FOIL method. This skill is a fundamental building block for more advanced algebraic concepts, making it an invaluable tool in your mathematical toolkit.

Checking Your Work

To ensure accuracy, it's always a good practice to check your work. One way to do this is by substituting a value for 'x' in both the original expression and the final simplified expression. If both expressions yield the same result, it's a strong indication that your multiplication and simplification were correct. For example, let's substitute x = 1 into the original expression (x+3)(x-9) and the simplified expression x² - 6x - 27:

  • Original Expression: (1+3)(1-9) = (4)(-8) = -32
  • Simplified Expression: (1)² - 6(1) - 27 = 1 - 6 - 27 = -32

Since both expressions evaluate to -32, we can be confident that our result is correct. This method of substitution provides a practical way to verify your algebraic manipulations and catch any potential errors. It's a valuable habit to develop, especially when dealing with more complex expressions and equations.

Conclusion

In this guide, we've explored the process of multiplying binomials, focusing on the example (x+3)(x-9). We've covered the basics of binomials, the significance of binomial multiplication in mathematics, and the step-by-step application of the FOIL method. We've also emphasized the importance of combining like terms for simplification and checking your work for accuracy. Mastering binomial multiplication is a crucial skill in algebra, serving as a foundation for more advanced mathematical concepts. By understanding and practicing these techniques, you'll be well-equipped to tackle a wide range of algebraic problems with confidence and precision. Remember, practice makes perfect, so keep working on similar problems to solidify your understanding.

For further learning and practice, you might find resources on websites like Khan Academy's Algebra I section particularly helpful.