Expanding (4x^5 + 11)^2 A Step-by-Step Guide

This article delves into the process of expanding the expression (4x^5 + 11)^2, providing a step-by-step guide to arrive at the correct equivalent expression. We will explore the fundamental algebraic principles involved, ensuring a clear understanding of the solution. This is a common type of problem encountered in algebra, particularly when dealing with polynomials and their expansions. Mastering this skill is crucial for success in higher-level mathematics. We will examine common pitfalls and how to avoid them, ensuring a solid grasp of the concepts. The correct identification and application of algebraic identities are key to simplifying and solving such expressions efficiently. By the end of this guide, you'll not only know the answer but also understand the underlying mathematical principles.

Unveiling the Expansion Process

The problem at hand requires us to expand the expression (4x^5 + 11)^2. This is a classic example of squaring a binomial, which can be efficiently done using the algebraic identity: (a + b)^2 = a^2 + 2ab + b^2. Let's break down how this applies to our specific problem. We identify 'a' as 4x^5 and 'b' as 11. Now, we can substitute these values into the identity. First, we need to square 'a', which means (4x⁵⁾2. Remember, when you raise a product to a power, you raise each factor to that power. So, (4x⁵⁾2 becomes 4^2 * (x⁵⁾2. This simplifies to 16x^10, as we multiply the exponents when raising a power to another power. Next, we need to calculate 2ab. This means 2 * (4x^5) * 11. Multiplying the coefficients, we get 2 * 4 * 11 = 88. So, this term becomes 88x^5. Finally, we need to square 'b', which is 11^2, resulting in 121. Now, we combine all these terms together according to the identity: a^2 + 2ab + b^2. This gives us 16x^10 + 88x^5 + 121. This is the expanded form of the given expression. This process highlights the importance of correctly applying algebraic identities to simplify and solve expressions. Understanding these identities allows for a more efficient and accurate approach to problem-solving.

Step-by-Step Solution

To solve the problem, we methodically apply the binomial expansion formula. The given expression is (4x^5 + 11)^2. We recognize this as a binomial squared, fitting the pattern (a + b)^2. The expansion formula for this pattern is a^2 + 2ab + b^2. Now, let's identify 'a' and 'b' in our expression. Here, 'a' is 4x^5 and 'b' is 11. We begin by calculating a^2. This means squaring 4x^5, which is (4x⁵⁾2. Remember the rules of exponents: when you raise a product to a power, you raise each factor to that power. Thus, (4x⁵⁾2 becomes 4^2 * (x⁵⁾2. Calculating 4^2 gives us 16. When raising a power to another power, we multiply the exponents, so (x⁵⁾2 becomes x^(5*2) = x^10. Therefore, a^2 is 16x^10. Next, we calculate 2ab. This is 2 * (4x^5) * 11. Multiplying the coefficients, we have 2 * 4 * 11 = 88. So, 2ab is 88x^5. Finally, we calculate b^2. This is 11^2, which equals 121. Now, we combine the results: a^2 + 2ab + b^2 = 16x^10 + 88x^5 + 121. This is the expanded form of the original expression. This step-by-step breakdown illustrates the importance of precision and attention to detail when applying algebraic formulas. Each step, from identifying 'a' and 'b' to correctly applying the exponent rules, is crucial for arriving at the correct answer. This methodical approach not only helps in solving the problem but also reinforces the understanding of the underlying principles.

Analyzing the Answer Choices

Now that we have expanded the expression (4x^5 + 11)^2 and arrived at 16x^10 + 88x^5 + 121, let's analyze the given answer choices to identify the correct one. Option A, 16x^5 + 121, is incorrect because it misses the crucial middle term (88x^5) that arises from the 2ab part of the binomial expansion. It also incorrectly retains the x term as x^5 instead of x^10. Option B, 16x^10 + 121, is also incorrect for the same reason. It correctly calculates the a^2 and b^2 terms but omits the 2ab term, leading to an incomplete expansion. Option C, 16x^10 + 88x^5 + 121, matches our calculated expansion perfectly. It includes all three terms – a^2, 2ab, and b^2 – with the correct coefficients and exponents. Therefore, this is the correct answer. Option D, 16x^25 + 88x^5 + 121, is incorrect because it incorrectly calculates the exponent of the first term. It seems to have multiplied the exponents 5 and 2 instead of applying the power of a power rule correctly. The correct application of the power of a power rule results in x^10, not x^25. This analysis highlights the importance of carefully comparing the expanded expression with the answer choices. It also demonstrates how understanding the binomial expansion formula and the rules of exponents is crucial for correctly identifying the equivalent expression. By methodically evaluating each option, we can confidently select the correct answer.

Common Mistakes and How to Avoid Them

When expanding expressions like (4x^5 + 11)^2, several common mistakes can occur. Understanding these pitfalls is crucial for accuracy. One frequent error is forgetting the middle term in the binomial expansion. The binomial expansion formula (a + b)^2 = a^2 + 2ab + b^2 includes the '2ab' term, which represents twice the product of 'a' and 'b'. Many students mistakenly expand the expression as just a^2 + b^2, neglecting this critical component. To avoid this, always remember the complete formula and consciously calculate the 2ab term. Another common mistake involves incorrectly applying the power of a power rule. When raising a term with an exponent to another power, like (x⁵⁾2, you multiply the exponents. A mistake here would be to add the exponents or not apply the power to the coefficient. Remember that (4x⁵⁾2 is 4^2 * (x⁵⁾2, which simplifies to 16x^10. A third common error is mishandling the coefficients. When calculating 2ab, ensure you multiply all the coefficients correctly. For example, in 2 * (4x^5) * 11, the coefficients 2, 4, and 11 must be multiplied together to get 88. Overlooking or miscalculating these multiplications can lead to an incorrect answer. To avoid these mistakes, practice careful and methodical expansion. Double-check each step, paying close attention to the application of exponent rules and the multiplication of coefficients. Using the binomial expansion formula correctly and being mindful of these common errors will significantly improve accuracy.

Conclusion

In conclusion, the expression equivalent to (4x^5 + 11)^2 is 16x^10 + 88x^5 + 121, which corresponds to answer choice C. This result is obtained by correctly applying the binomial expansion formula (a + b)^2 = a^2 + 2ab + b^2, where a = 4x^5 and b = 11. The expansion involves squaring 'a' to get 16x^10, calculating 2ab to get 88x^5, and squaring 'b' to get 121. The sum of these terms gives the final expanded expression. Understanding the binomial expansion formula is crucial for solving this type of problem efficiently and accurately. This skill is fundamental in algebra and has applications in various mathematical contexts. Avoiding common mistakes, such as neglecting the 2ab term or misapplying exponent rules, is essential for success. By practicing these types of expansions and reinforcing the underlying algebraic principles, one can confidently tackle similar problems. This comprehensive guide has not only provided the solution but also elucidated the step-by-step process, common pitfalls, and strategies to ensure a thorough understanding of the concept. Mastering these skills will undoubtedly contribute to a stronger foundation in mathematics.