Select The Correct Answer.Which Expression Is Equivalent To The Given Polynomial Expression? \left(9v^4 + 2\right) + V^2\left(v^2w^2 + 2w^3 - 2v^2\right) - \left(-13v^2w^3 + 7v^4\right ]A. 14 V 4 + V 4 W 2 + 15 V 2 W 3 + 2 14v^4 + V^4w^2 + 15v^2w^3 + 2 14 V 4 + V 4 W 2 + 15 V 2 W 3 + 2 B. $v 4w 2

Introduction

Polynomial expressions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will explore how to simplify a given polynomial expression by combining like terms and applying the distributive property. We will also examine a specific example and determine which expression is equivalent to the given polynomial expression.

Understanding Polynomial Expressions

A polynomial expression is a mathematical expression that consists of variables, coefficients, and exponents. It can be written in the form:

$$a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$$

where $a_n, a_{n-1}, \ldots, a_1, a_0$ are coefficients, $x$ is the variable, and $n$ is the degree of the polynomial.

Simplifying Polynomial Expressions

To simplify a polynomial expression, we need to combine like terms and apply the distributive property. Like terms are terms that have the same variable and exponent. We can combine like terms by adding or subtracting their coefficients.

Example: Simplifying a Polynomial Expression

Let's consider the following polynomial expression:

$$\left(9v^4 + 2\right) + v^2\left(v^2w^2 + 2w^3 - 2v^2\right) - \left(-13v^2w^3 + 7v^4\right)$$

Our goal is to simplify this expression by combining like terms and applying the distributive property.

Step 1: Distribute the $v^2$ Term

First, we need to distribute the $v^2$ term to the terms inside the parentheses:

$$v^2\left(v^2w^2 + 2w^3 - 2v^2\right) = v^4w^2 + 2v^2w^3 - 2v^4$$

Step 2: Combine Like Terms

Next, we need to combine like terms:

$$9v^4 + 2v^4 - 2v^4 = 9v^4$$

$$2v^2w^3 + 2v^2w^3 = 4v^2w^3$$

Step 3: Simplify the Expression

Now, we can simplify the expression by combining the like terms:

$$\left(9v^4 + 2\right) + v^2\left(v^2w^2 + 2w^3 - 2v^2\right) - \left(-13v^2w^3 + 7v^4\right)$$

$$= 9v^4 + 2 + v^4w^2 + 2v^2w^3 - 2v^4 + 13v^2w^3 - 7v^4$$

$$= 9v^4 - 7v^4 - 2v^4 + v^4w^2 + 2v^2w^3 + 13v^2w^3$$

$$= v^4w^2 + 15v^2w^3 + 0v^4 + 2$$

Conclusion

In conclusion, the simplified expression is:

$$v^4w^2 + 15v^2w^3 + 2$$

This expression is equivalent to the given polynomial expression.

Answer

The correct answer is:

A. $14v^4 + v^4w^2 + 15v^2w^3 + 2$

However, we have found that the correct simplified expression is:

$v^4w^2 + 15v^2w^3 + 2$

This expression is not among the options provided. Therefore, we cannot select the correct answer from the options provided.

Discussion

This example illustrates the importance of carefully simplifying polynomial expressions by combining like terms and applying the distributive property. It also highlights the need to be cautious when selecting the correct answer from a list of options, as the correct answer may not be among the options provided.

Final Thoughts

Introduction

In our previous article, we explored how to simplify polynomial expressions by combining like terms and applying the distributive property. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in simplifying polynomial expressions.

Q: What is a polynomial expression?

A: A polynomial expression is a mathematical expression that consists of variables, coefficients, and exponents. It can be written in the form:

$$a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$$

where $a_n, a_{n-1}, \ldots, a_1, a_0$ are coefficients, $x$ is the variable, and $n$ is the degree of the polynomial.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that the product of a sum is equal to the sum of the products. In other words, for any numbers $a, b, c$:

$$a(b + c) = ab + ac$$

Q: How do I simplify a polynomial expression?

A: To simplify a polynomial expression, you need to combine like terms and apply the distributive property. Like terms are terms that have the same variable and exponent. You can combine like terms by adding or subtracting their coefficients.

Q: What is the difference between a like term and a unlike term?

A: A like term is a term that has the same variable and exponent. For example, $2x^2$ and $3x^2$ are like terms because they have the same variable ($x$) and exponent ($2$). A unlike term is a term that has a different variable or exponent. For example, $2x^2$ and $3y^2$ are unlike terms because they have different variables ($x$ and $y$).

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to multiply each term inside the parentheses by the term outside the parentheses. For example, if you have the expression $(2x + 3)(4x + 1)$, you would multiply each term inside the parentheses by the term outside the parentheses:

$$(2x + 3)(4x + 1) = 2x(4x + 1) + 3(4x + 1)$$

$$= 8x^2 + 2x + 12x + 3$$

$$= 8x^2 + 14x + 3$$

Q: What is the difference between a polynomial expression and an algebraic expression?

A: A polynomial expression is a mathematical expression that consists of variables, coefficients, and exponents. An algebraic expression is a mathematical expression that consists of variables, coefficients, and constants. For example, $2x + 3$ is an algebraic expression, but $2x^2 + 3x + 1$ is a polynomial expression.

Q: How do I determine the degree of a polynomial expression?

A: The degree of a polynomial expression is the highest exponent of the variable. For example, in the expression $2x^2 + 3x + 1$, the highest exponent of the variable $x$ is $2$, so the degree of the expression is $2$.

Conclusion

In conclusion, simplifying polynomial expressions is a crucial skill for any math enthusiast. By combining like terms and applying the distributive property, we can simplify complex expressions and arrive at the correct solution. We hope that this Q&A guide has provided a clear and concise overview of the concepts and techniques involved in simplifying polynomial expressions.

Final Thoughts

In conclusion, simplifying polynomial expressions is a fundamental concept in algebra that requires a deep understanding of the distributive property and the ability to combine like terms. By mastering these skills, you will be able to simplify complex expressions and arrive at the correct solution. We hope that this Q&A guide has provided a helpful resource for you to learn and practice simplifying polynomial expressions.