Expanding (x + 7)³ A Step-by-Step Guide
In this comprehensive guide, we will delve into the process of expanding the function f(x) = (x + 7)³. This is a fundamental concept in algebra, often encountered in various mathematical contexts, including polynomial expansions, calculus, and equation solving. Understanding how to expand such functions is crucial for simplifying expressions, finding roots, and performing other algebraic manipulations. This article aims to provide a detailed, step-by-step explanation of the expansion process, ensuring a clear understanding for readers of all backgrounds. We will explore different methods, highlight key concepts, and provide examples to illustrate the techniques involved. Whether you are a student learning algebra for the first time or someone looking to refresh your knowledge, this guide will equip you with the necessary skills to confidently expand cubic functions.
Understanding the Basics of Polynomial Expansion
Before we dive into expanding f(x) = (x + 7)³, it's essential to grasp the fundamental principles of polynomial expansion. Polynomial expansion involves multiplying out expressions containing variables and constants raised to various powers. The key to successfully expanding polynomials lies in the distributive property, which states that a(b + c) = ab + ac. This property allows us to multiply each term within one set of parentheses by each term within another set. When dealing with cubic functions like (x + 7)³, we are essentially multiplying the expression (x + 7) by itself three times. This can be a bit more complex than expanding quadratic expressions, but the underlying principle remains the same. We need to systematically multiply each term to ensure we account for all possible combinations. Furthermore, understanding the binomial theorem can provide a shortcut for expanding expressions of the form (a + b)ⁿ, where n is a positive integer. The binomial theorem provides a formula for directly calculating the coefficients in the expansion, which can save time and reduce the chances of making errors. In the following sections, we will explore both the step-by-step multiplication method and how the binomial theorem can be applied to expand f(x) = (x + 7)³.
Method 1: Step-by-Step Multiplication
The most straightforward method to expand f(x) = (x + 7)³ is through step-by-step multiplication. This involves first multiplying (x + 7) by itself, and then multiplying the result by (x + 7) again. Let's break this down: First, we multiply (x + 7) by (x + 7). This can be visualized as: (x + 7)(x + 7). Applying the distributive property (often remembered by the acronym FOIL - First, Outer, Inner, Last), we get: x times x equals x², x times 7 equals 7x, 7 times x equals 7x, and 7 times 7 equals 49. Combining these terms, we have x² + 7x + 7x + 49, which simplifies to x² + 14x + 49. Now, we need to multiply this quadratic expression by (x + 7). This means we are multiplying (x² + 14x + 49) by (x + 7). Again, we apply the distributive property, ensuring that each term in the quadratic expression is multiplied by both x and 7. This gives us: x(x² + 14x + 49) + 7*(x² + 14x + 49)*. Distributing the x, we get x³ + 14x² + 49x. Distributing the 7, we get 7x² + 98x + 343. Now, we combine these two expressions: (x³ + 14x² + 49x) + (7x² + 98x + 343). Finally, we combine like terms: x³ remains as is, 14x² + 7x² equals 21x², 49x + 98x equals 147x, and 343 remains as is. Thus, the expanded form of (x + 7)³ is x³ + 21x² + 147x + 343. This method, while slightly lengthy, is reliable and helps to visualize the expansion process. It reinforces the understanding of the distributive property and ensures accuracy by breaking down the problem into smaller, manageable steps.
Method 2: Using the Binomial Theorem
Another powerful method for expanding (x + 7)³ is by utilizing the binomial theorem. The binomial theorem provides a formula for expanding expressions of the form (a + b)ⁿ, where n is a positive integer. The theorem states that: (a + b)ⁿ = Σ [nCk * a^(n-k) * b^k], where k ranges from 0 to n, and nCk represents the binomial coefficient, often read as "n choose k." The binomial coefficient can be calculated using the formula: nCk = n! / (k!(n-k)!), where n! (n factorial) is the product of all positive integers up to n. For (x + 7)³, a = x, b = 7, and n = 3. Applying the binomial theorem, we need to calculate the coefficients for k = 0, 1, 2, and 3. For k = 0, 3C0 = 3! / (0!3!) = 1. For k = 1, 3C1 = 3! / (1!2!) = 3. For k = 2, 3C2 = 3! / (2!1!) = 3. For k = 3, 3C3 = 3! / (3!0!) = 1. Now we can plug these coefficients into the binomial theorem formula: For k = 0: 1 * x^(3-0) * 7^0 = 1 * x³ * 1 = x³. For k = 1: 3 * x^(3-1) * 7^1 = 3 * x² * 7 = 21x². For k = 2: 3 * x^(3-2) * 7^2 = 3 * x * 49 = 147x. For k = 3: 1 * x^(3-3) * 7^3 = 1 * 1 * 343 = 343. Summing these terms together, we get x³ + 21x² + 147x + 343, which is the same result we obtained using the step-by-step multiplication method. The binomial theorem provides a more direct route to the solution, especially for higher powers, as it avoids the need for repeated multiplication. However, it requires understanding the concepts of binomial coefficients and factorials.
Filling in the Blanks: Completing the Expansion
Now that we have successfully expanded f(x) = (x + 7)³ using two different methods, we can confidently fill in the blanks in the original expression. The expanded form is x³ + 21x² + 147x + 343. Comparing this with the provided format: 1x³ + [?]x² + []x + ▢, we can identify the missing coefficients. The coefficient of x² is 21, the coefficient of x is 147, and the constant term is 343. Therefore, the completed expression is: 1x³ + 21x² + 147x + 343. This exercise highlights the importance of accurately expanding the function to determine the correct coefficients. It also reinforces the understanding of polynomial structure and the roles of coefficients and constants within the expression.
Common Mistakes to Avoid
When expanding functions like f(x) = (x + 7)³, it's crucial to be aware of common mistakes that can occur. One of the most frequent errors is misapplying the distributive property. This often happens when terms are missed during the multiplication process, leading to an incomplete expansion. To avoid this, it's helpful to be systematic and ensure that each term in one set of parentheses is multiplied by every term in the other set. Another common mistake is incorrectly combining like terms. This can occur if the signs are not carefully tracked or if terms with different powers of x are mistakenly combined. Double-checking the signs and ensuring that only terms with the same power of x are added or subtracted is essential. When using the binomial theorem, errors can arise if the binomial coefficients are calculated incorrectly or if the formula is misapplied. It's important to understand the binomial coefficient formula and to carefully substitute the values of n and k. Additionally, using Pascal's Triangle as a quick reference for binomial coefficients can be a helpful strategy. Another error is forgetting the order of operations (PEMDAS/BODMAS), especially when simplifying the expression after expansion. Exponents should be dealt with before multiplication and division, and addition and subtraction should be performed last. By being mindful of these common mistakes and taking a careful, step-by-step approach, you can significantly reduce the likelihood of errors and ensure accurate expansion of polynomial functions.
Conclusion: Mastering Polynomial Expansion
In conclusion, expanding the function f(x) = (x + 7)³ is a fundamental skill in algebra that can be mastered through a clear understanding of the underlying principles and careful application of the appropriate methods. We explored two primary approaches: step-by-step multiplication and the binomial theorem. The step-by-step method, while more laborious, provides a solid grasp of the distributive property and the mechanics of polynomial multiplication. The binomial theorem offers a more direct and efficient route, especially for higher powers, but requires familiarity with binomial coefficients and factorials. By understanding both methods, you can choose the one that best suits the specific problem and your personal preference. We also highlighted common mistakes to avoid, emphasizing the importance of careful application of the distributive property, accurate calculation of coefficients, and systematic combination of like terms. The ability to expand polynomial functions is not only essential for solving algebraic equations but also forms the basis for more advanced topics in calculus and other areas of mathematics. Therefore, investing time in mastering this skill will pay dividends in your future mathematical studies. Remember to practice regularly and to double-check your work to ensure accuracy. With consistent effort, you can confidently expand a wide range of polynomial expressions and apply this knowledge to solve complex problems.
