Simplifying Polynomial Expressions: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into the world of polynomial expressions. Specifically, we're tackling the question: "Which expression is equivalent to this polynomial expression? $\left(2 x^5+3 y^4\right)\left(-4 x^2+9 y^4\right)$" Don't worry if it looks a bit intimidating at first – we'll break it down step by step and make sure you understand the process. This is a common type of problem in algebra, and mastering it will give you a solid foundation for more complex mathematical concepts.

Understanding the Problem: Polynomial Multiplication

At its core, this problem involves polynomial multiplication. We have two binomials (expressions with two terms) multiplied together. Our goal is to expand this product into a single, simplified polynomial. The key to solving this lies in the distributive property – which states that you multiply each term in the first binomial by each term in the second binomial. It’s like spreading out the love (or in this case, the multiplication) to all the terms!

In our case, we have $\left(2 x^5+3 y^4\right)\left(-4 x^2+9 y^4\right)$. Let's visualize how we're going to use the distributive property. We will multiply each term of the first binomial $(2x^5 + 3y^4)$ with each term of the second binomial $(-4x^2 + 9y^4)$. This means we will perform the following multiplications:

• $2x^5$ by $-4x^2$
• $2x^5$ by $9y^4$
• $3y^4$ by $-4x^2$
• $3y^4$ by $9y^4$

Once we perform these multiplications, we will combine the like terms to get the final answer. Now, let’s go through this step by step. Remember, the goal is to make it clear and simple, ensuring that you can follow the process without any confusion. So, let’s begin!

Step-by-Step Solution: Expanding and Simplifying

Alright, let’s get our hands dirty and start expanding the expression. We will multiply each term in the first parenthesis by each term in the second parenthesis. This is where the distributive property comes into play. Pay close attention to the signs and exponents!

1. Multiply $2x^5$ by $-4x^2$:

   • $2x^5 * -4x^2 = -8x^{(5+2)} = -8x^7$
   • Remember, when multiplying terms with exponents, you add the exponents. The coefficient 2 is multiplied by -4 to get -8.

2. Multiply $2x^5$ by $9y^4$:

   • $2x^5 * 9y^4 = 18x^5y^4$
   • Here, we multiply the coefficients and simply write down the variables with their exponents, as they are different.

3. Multiply $3y^4$ by $-4x^2$:

   • $3y^4 * -4x^2 = -12x^2y^4$
   • Similar to the previous step, we multiply the coefficients (3 and -4) and write down the variables.

4. Multiply $3y^4$ by $9y^4$:

   • $3y^4 * 9y^4 = 27y^{(4+4)} = 27y^8$
   • Again, we multiply the coefficients and add the exponents of the same variable (y).

Now, we have the following terms: $-8x^7$, $18x^5y^4$, $-12x^2y^4$, and $27y^8$. The next step is to write down the expanded expression by combining these terms.

The expanded expression becomes: $-8x^7 + 18x^5y^4 - 12x^2y^4 + 27y^8$. At this stage, it's very important to ensure all terms have been accounted for and the calculations are correct. If you find it helpful, you can even double-check the individual multiplications before moving on to the next step. So far, we've successfully expanded the original expression using the distributive property. The result is a collection of terms that we can then simplify, if possible.

Identifying Like Terms and Simplifying

In this particular expanded expression, there are no like terms. Like terms are terms that have the same variables raised to the same powers. For example, $3x^2$ and $5x^2$ are like terms, but $3x^2$ and $3x^3$ are not. In our expanded form, each term has a unique combination of variables and exponents: $-8x^7$, $18x^5y^4$, $-12x^2y^4$, and $27y^8$. Because there are no like terms, we cannot simplify the expression further by combining any terms.

Therefore, the final simplified expression remains the same as our expanded form: $-8x^7 + 18x^5y^4 - 12x^2y^4 + 27y^8$. This result is the equivalent expression we were looking for, derived from our initial polynomial multiplication.

It’s important to practice recognizing like terms and knowing how to combine them because it’s a crucial skill in algebra. When you have like terms, you simply add or subtract their coefficients while keeping the variable part the same. This ability to simplify polynomial expressions is used extensively in solving equations, graphing functions, and in many applications of mathematics. Now that we have our final simplified expression, let’s move on to the next section where we identify the correct answer choice from the options provided.

Matching the Solution to the Answer Choices

Now that we've meticulously solved and simplified our polynomial expression, it's time to compare our result with the multiple-choice options. Our final answer is $-8x^7 + 18x^5y^4 - 12x^2y^4 + 27y^8$. We need to see which of the provided options matches this expression.

Let’s go through each of the answer choices:

A. $-2x^7 + 11x^5y^4 - x^2y^4 + 12y^8$: This option does not match our calculated result because the coefficients and variables do not align. B. $-8x^7 + 27y^8$: This option is missing the $x^5y^4$ and $x^2y^4$ terms, so it is incorrect. C. $-2x^{10} + 11x^5y^4 - x^2y^4 + 12y^{16}$: This option is completely different from our result, with incorrect exponents and coefficients. D. $-8x^7 + 18x^5y^4 - 12x^2y^4 + 27y^8$: This is the correct answer! It matches our calculated result perfectly in terms of coefficients, variables, and exponents.

Therefore, after comparing our simplified expression with the provided choices, the correct answer is D. This is a great example of how important it is to be precise in your calculations, as even a small mistake can lead to an incorrect answer. Always double-check your work, and don't hesitate to go back and review each step if you're not sure. Practicing similar problems will help you become more comfortable with polynomial multiplication and simplification.

Conclusion: Mastering Polynomial Expressions

Congratulations! You've successfully navigated the world of polynomial expressions, mastered polynomial multiplication, and found the correct answer. The key takeaways from this problem are:

1. Understanding the Distributive Property: This is the cornerstone of expanding polynomial expressions.
2. Accurate Multiplication of Terms: Pay close attention to the signs and exponents when multiplying.
3. Identifying and Combining Like Terms: This step ensures your expression is simplified correctly.
4. Comparing with Answer Choices: This is crucial to ensure you've selected the correct solution.

This method not only helps solve the given problem but also lays a strong foundation for tackling more complex algebraic concepts. Polynomials are a fundamental part of algebra and are used extensively in many areas of mathematics and science. By mastering these basics, you’re well on your way to success in your mathematical journey. So, keep practicing, keep learning, and don’t be afraid to take on new challenges. Every step you take, you are building a stronger understanding of mathematics. Keep up the great work, and happy solving!

For more detailed explanations and examples, you might find the following resources helpful:

• Khan Academy (https://www.khanacademy.org/) - Khan Academy has a wealth of free resources, including videos and practice exercises, that cover polynomial expressions and other related topics. You can find detailed lessons on multiplying polynomials and simplifying expressions.

• Your textbook or online educational resources, such as Math is Fun (https://www.mathsisfun.com/) can offer additional examples and practice problems to help solidify your understanding.

Keep practicing, and you will become a master of polynomial expressions in no time! Keep up the amazing work!