Select The Correct Answer.Which Expression Is Equivalent To This Polynomial Expression?$ \left(5xy^2 + 3x^2 - 7\right) + \left(3x 2y 2 - Xy^2 + 3y^2 + 4\right) $A. $ 9x 2y 2 + 4xy^2 - 3 $B. $ 3x 2y 2 + 6xy^2 + 6x^2 + 3 $C.

Introduction

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

What are Polynomial Expressions?

A polynomial expression is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. It can be written in the form:

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

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

Simplifying Polynomial Expressions

To simplify a polynomial expression, we need to combine like terms. Like terms are terms that have the same variable raised to the same power. For example, $3x^2$ and $2x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

Step 1: Identify Like Terms

The first step in simplifying a polynomial expression is to identify like terms. We need to look for terms that have the same variable raised to the same power.

Step 2: Combine Like Terms

Once we have identified like terms, we can combine them by adding or subtracting their coefficients. For example, if we have the terms $3x^2$ and $2x^2$, we can combine them by adding their coefficients:

$$3x^2 + 2x^2 = 5x^2$$

Step 3: Simplify the Expression

After combining like terms, we can simplify the expression by removing any unnecessary parentheses or brackets.

Example: Simplifying a Polynomial Expression

Let's consider the following polynomial expression:

$$\left(5xy^2 + 3x^2 - 7\right) + \left(3x^2y^2 - xy^2 + 3y^2 + 4\right)$$

To simplify this expression, we need to combine like terms. We can start by identifying like terms:

• $5xy^2$ and $-xy^2$ are like terms because they both have the variable $x$ raised to the power of 1 and the variable $y$ raised to the power of 2.
• $3x^2$ and $3x^2y^2$ are like terms because they both have the variable $x$ raised to the power of 2.
• $-7$ and $4$ are like terms because they are both constants.

Now, we can combine like terms:

• $5xy^2 + (-xy^2) = 4xy^2$
• $3x^2 + 3x^2y^2 = 3x^2(1 + y^2)$
• $-7 + 4 = -3$

So, the simplified expression is:

$$4xy^2 + 3x^2(1 + y^2) - 3$$

Which Expression is Equivalent to the Given Polynomial Expression?

Now, let's examine the answer choices and determine which expression is equivalent to the given polynomial expression.

A. $9x^2y^2 + 4xy^2 - 3$

B. $3x^2y^2 + 6xy^2 + 6x^2 + 3$

C. $3x^2y^2 + 6xy^2 + 6x^2 + 3$

We can see that option C is the correct answer because it matches the simplified expression we obtained earlier:

$$4xy^2 + 3x^2(1 + y^2) - 3$$

However, we can simplify option C further by combining like terms:

$$3x^2y^2 + 6xy^2 + 6x^2 + 3 = 3x^2y^2 + 6xy^2 + 6x^2 + 3 = 3x^2(y^2 + 2) + 6xy^2 + 3$$

This expression is equivalent to the given polynomial expression.

Conclusion

Q: What is a polynomial expression?

A: A polynomial expression is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. It can be written in the form:

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

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

Q: How do I simplify a polynomial expression?

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

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, $3x^2$ and $2x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

Q: How do I identify like terms?

A: To identify like terms, you need to look for terms that have the same variable raised to the same power. You can do this by examining the variables and their exponents in each term.

Q: Can I simplify a polynomial expression with variables and constants?

A: Yes, you can simplify a polynomial expression with variables and constants. You can combine like terms, including terms with variables and constants.

Q: What is the order of operations for simplifying polynomial expressions?

A: The order of operations for simplifying polynomial expressions is:

1. Identify like terms
2. Combine like terms
3. Simplify the expression

Q: Can I use a calculator to simplify polynomial expressions?

A: Yes, you can use a calculator to simplify polynomial expressions. However, it's always a good idea to check your work by hand to make sure you understand the process.

Q: How do I know if a polynomial expression is simplified?

A: A polynomial expression is simplified when there are no like terms left to combine. You can check this by looking for terms with the same variable raised to the same power.

Q: Can I simplify a polynomial expression with negative coefficients?

A: Yes, you can simplify a polynomial expression with negative coefficients. You can combine like terms, including terms with negative coefficients.

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

A: A polynomial expression is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. An algebraic expression is a more general term that includes polynomial expressions, as well as other types of expressions.

Q: Can I simplify a polynomial expression with fractional coefficients?

A: Yes, you can simplify a polynomial expression with fractional coefficients. You can combine like terms, including terms with fractional coefficients.

Q: How do I know if a polynomial expression is equivalent to another expression?

A: Two polynomial expressions are equivalent if they have the same value for all possible values of the variables. You can check this by simplifying both expressions and comparing the results.

Q: Can I use a graphing calculator to simplify polynomial expressions?

A: Yes, you can use a graphing calculator to simplify polynomial expressions. However, it's always a good idea to check your work by hand to make sure you understand the process.

Conclusion

In conclusion, simplifying polynomial expressions is a crucial skill to master in algebra. By combining like terms, you can simplify complex expressions and make them easier to work with. We hope this Q&A article has helped you understand the process of simplifying polynomial expressions and has provided you with the tools you need to succeed in algebra.