Expanding 2a(a^2 + 3) A Step-by-Step Solution

Expanding algebraic expressions is a fundamental skill in mathematics. In this comprehensive guide, we will explore the process of expanding the expression $2a(a^2 + 3)$, providing a step-by-step solution and a detailed explanation. Our goal is to make this concept clear and accessible to learners of all levels, ensuring that you not only understand the solution but also the underlying principles. Understanding how to expand expressions like $2a(a^2 + 3)$ is crucial for simplifying complex equations, solving for unknowns, and building a strong foundation in algebra. By mastering this skill, you'll be well-equipped to tackle more advanced mathematical problems. In this article, we'll break down the expansion process, discuss common mistakes to avoid, and provide practice examples to solidify your understanding. Let's begin by understanding the distributive property, which is the cornerstone of expanding such expressions. The distributive property states that $a(b + c) = ab + ac$. This means that we multiply the term outside the parentheses by each term inside the parentheses. Applying this to our expression, we have $2a(a^2 + 3)$. This means we need to multiply $2a$ by both $a^2$ and $3$. So, let’s proceed step by step to solve this.

Step-by-Step Expansion of $2a(a^2 + 3)$

The expansion of the expression $2a(a^2 + 3)$ involves applying the distributive property. Here’s a detailed breakdown of each step:

Step 1: Distribute $2a$ to $a^2$

The first step in expanding the expression is to multiply $2a$ by the first term inside the parentheses, which is $a^2$. When multiplying terms with exponents, we add the exponents. In this case, we have:

$2a * a^2$

Here, $2a$ can be thought of as $2a^1$. So, we are multiplying $2a^1$ by $a^2$. The rule for multiplying exponents states that $a^m * a^n = a^{m+n}$. Applying this rule, we get:

$2a^1 * a^2 = 2a^{1+2} = 2a^3$

Therefore, the result of multiplying $2a$ by $a^2$ is $2a^3$. This is a critical step, as it sets the foundation for the rest of the expansion. Understanding the rule of adding exponents is essential for correctly multiplying these terms. Make sure to keep the coefficient (the number in front of the variable) in mind as you perform this multiplication. The coefficient 2 in $2a$ plays a crucial role in the final result. Misunderstanding or overlooking this coefficient can lead to incorrect answers. Now, we proceed to the next part of the expansion.

Step 2: Distribute $2a$ to $3$

The next step in expanding the expression is to multiply $2a$ by the second term inside the parentheses, which is $3$. This is a straightforward multiplication:

$2a * 3$

When multiplying a term with a variable by a constant, we simply multiply the coefficients. In this case, we multiply $2$ by $3$:

$2 * 3 = 6$

So, the result of multiplying $2a$ by $3$ is $6a$. This step is often easier than the previous one, but it’s still important to perform it accurately. Make sure you understand that we are multiplying the coefficient of $a$ (which is 2) by the constant 3. The variable $a$ remains as it is, as there are no other $a$ terms to combine with in this multiplication. It’s also worth noting that this step highlights the importance of the distributive property, which ensures that we multiply the term outside the parentheses by every term inside. Skipping this step or making a mistake here would lead to an incorrect expansion of the original expression. With both parts of the distribution completed, we can now combine the results.

Step 3: Combine the Results

Now that we have multiplied $2a$ by both terms inside the parentheses, we can combine the results. From Step 1, we found that $2a * a^2 = 2a^3$, and from Step 2, we found that $2a * 3 = 6a$. So, we add these two results together:

$2a^3 + 6a$

This is the expanded form of the original expression. There are no like terms to combine further, so this is our final answer. Combining the results is a crucial step because it completes the expansion process. It’s important to double-check that you have correctly added the results from each multiplication. In this case, $2a^3$ and $6a$ are not like terms because they have different exponents for the variable $a$. Therefore, they cannot be combined any further. Recognizing and understanding like terms is an important skill in algebra, and it helps prevent mistakes in simplification. The final expression, $2a^3 + 6a$, represents the fully expanded form of $2a(a^2 + 3)$. To solidify your understanding, it's beneficial to review the entire process and perhaps try similar examples on your own. This will reinforce the concept and improve your confidence in expanding algebraic expressions.

Final Answer and Conclusion

After expanding the expression $2a(a^2 + 3)$, we found the result to be:

$2a^3 + 6a$

Therefore, the correct answer is option B, which is $2a^3 + 6a$. This comprehensive guide has walked you through each step of the expansion process, emphasizing the distributive property and the rules of exponents. By understanding these principles, you can confidently expand similar algebraic expressions. Remember, practice is key to mastering these skills. Try expanding other expressions and double-checking your work to reinforce your understanding. The ability to expand algebraic expressions is a foundational skill in mathematics, and mastering it will undoubtedly help you in more advanced topics. If you encounter any difficulties, revisit the steps outlined in this guide and consider working through additional examples. With practice and patience, you can become proficient in expanding algebraic expressions. Always remember to apply the distributive property correctly and pay close attention to the rules of exponents. These are the key elements to successfully expanding expressions like $2a(a^2 + 3)$. Understanding these concepts will not only help you solve mathematical problems but also enhance your overall problem-solving skills. Keep practicing, and you'll see significant improvement in your algebraic abilities.