Equivalent Polynomial Expression: Step-by-Step Solution

In this article, we'll break down how to find the equivalent expression for the polynomial: (9v⁴ + 2) + v²(v²w² + 2w³ - 2v²) - (-13v²w³ + 7v⁴). Polynomial expressions might seem daunting at first, but with a systematic approach, they become much easier to handle. We'll go through each step, explaining the logic behind it, so you can confidently tackle similar problems in the future. Let's dive in and simplify this expression!

Understanding Polynomial Expressions

Before we jump into solving the problem, let's define what a polynomial expression is. A polynomial is an expression consisting of variables (like v and w in our case) and coefficients, combined using addition, subtraction, and non-negative integer exponents. Examples of polynomials include 3x² + 2x - 1, 5y⁴ - 7y + 2, and the expression we're tackling today. Simplifying polynomial expressions often involves distributing, combining like terms, and rearranging to arrive at a more concise form. The goal is to rewrite the expression in a way that's easier to understand and work with. Remember, the key to simplifying complex expressions is to break them down into smaller, manageable steps. This makes the entire process less overwhelming and helps prevent errors. So, let's keep this in mind as we proceed through each step of solving our polynomial expression.

When dealing with polynomials, it’s crucial to understand the order of operations (PEMDAS/BODMAS) and the rules for combining like terms. Distributing terms correctly and identifying like terms are fundamental skills that make polynomial manipulation much smoother. A like term is a term that has the same variables raised to the same powers. For example, 3x² and -5x² are like terms, while 3x² and 3x³ are not. Combining like terms involves adding or subtracting their coefficients while keeping the variable part the same. Polynomial expressions appear frequently in algebra and calculus, so mastering the techniques for simplifying them is essential for success in these areas. Understanding the structure and properties of polynomials not only helps in solving algebraic equations but also in various applications, such as curve fitting, optimization problems, and modeling physical phenomena.

Step-by-Step Simplification

Our main task is to determine which expression is equivalent to: (9v⁴ + 2) + v²(v²w² + 2w³ - 2v²) - (-13v²w³ + 7v⁴). To begin, we need to distribute the v² term inside the parentheses. This means we multiply v² by each term inside the parenthesis: v² * v²w² = v⁴w², v² * 2w³ = 2v²w³, and v² * -2v² = -2v⁴. After distribution, our expression looks like this:

(9v⁴ + 2) + v⁴w² + 2v²w³ - 2v⁴ - (-13v²w³ + 7v⁴)

Next, we need to address the subtraction of the expression in the last set of parentheses. Subtracting a negative term is the same as adding its positive counterpart. So, we distribute the negative sign across the terms inside the parentheses, changing the signs of each term: -(-13v²w³) = +13v²w³, and -(7v⁴) = -7v⁴. Now, our expression is:

(9v⁴ + 2) + v⁴w² + 2v²w³ - 2v⁴ + 13v²w³ - 7v⁴

Combining Like Terms

Now comes the critical step: identifying and combining like terms. Remember, like terms have the same variables raised to the same powers. In our expression, we have several terms with v⁴ and v²w³. Let’s group them together:

v⁴ terms: 9v⁴, -2v⁴, -7v⁴ v²w³ terms: 2v²w³, 13v²w³

We also have the v⁴w² term and the constant term +2, which don’t have any like terms to combine with. Now, let's combine the v⁴ terms: 9v⁴ - 2v⁴ - 7v⁴. This simplifies to (9 - 2 - 7)v⁴ = 0v⁴ = 0. So, the v⁴ terms cancel out. Next, we combine the v²w³ terms: 2v²w³ + 13v²w³. This simplifies to (2 + 13)v²w³ = 15v²w³. We now have simplified our expression significantly. The remaining terms are v⁴w² and the constant +2. So, after combining all like terms, our simplified expression is:

v⁴w² + 15v²w³ + 2

This process of identifying and combining like terms is a cornerstone of simplifying polynomial expressions. It allows us to reduce the complexity of the expression and make it more manageable. By paying close attention to the variables and their exponents, we can accurately combine terms and arrive at the simplest form of the polynomial. In the next section, we'll compare our simplified expression with the given options and choose the correct answer.

Matching the Equivalent Expression

After simplifying the given polynomial expression, we arrived at: v⁴w² + 15v²w³ + 2. Now, let's compare this result with the answer choices provided to determine which one matches our simplified expression. The given options are:

A. v⁴w² + 15v²w³ + 2 B. 14v⁴ + v⁴w² + 15v²w³ + 2 C. 16v⁶w⁵ + 2 D. 14v⁴ + 16

By direct comparison, we can see that option A, v⁴w² + 15v²w³ + 2, exactly matches our simplified expression. This indicates that option A is the correct equivalent expression for the given polynomial. The other options contain terms or coefficients that do not align with our result, making them incorrect. For instance, option B includes a 14v⁴ term, which we eliminated during our simplification process. Option C presents a completely different set of terms (16v⁶w⁵), and option D includes a 14v⁴ term and lacks the v⁴w² and v²w³ terms. Therefore, the process of simplifying and comparing allows us to confidently select the correct equivalent expression. It’s always a good practice to double-check each term and coefficient to ensure an accurate match. In this case, the match is clear, confirming that our step-by-step simplification led us to the correct answer.

Conclusion

In summary, we've successfully simplified the polynomial expression (9v⁴ + 2) + v²(v²w² + 2w³ - 2v²) - (-13v²w³ + 7v⁴). By carefully distributing, combining like terms, and comparing the result with the given options, we found that the equivalent expression is v⁴w² + 15v²w³ + 2 (Option A). This exercise highlights the importance of a methodical approach to solving polynomial expressions. By breaking down complex problems into smaller, manageable steps, we can confidently arrive at the correct solution. Remember, practice is key to mastering these skills. The more you work with polynomial expressions, the more comfortable and proficient you'll become in simplifying them.

If you're interested in learning more about polynomials and algebraic expressions, you can explore resources like Khan Academy's Algebra I section, which offers comprehensive lessons and practice exercises.