Select The Correct Answer.Which Expression Is Equivalent To This Polynomial Expression? \left(8x^2y^2 - 9x^2y + 9y\right) - \left(6x^2y - Xy^2 + 4y\right ]A. − 7 X 2 Y 2 − X Y 2 + 13 Y -7x^2y^2 - Xy^2 + 13y − 7 X 2 Y 2 − X Y 2 + 13 Y B. 8 X 2 Y 2 − 15 X 2 Y + X Y 2 + 5 Y 8x^2y^2 - 15x^2y + Xy^2 + 5y 8 X 2 Y 2 − 15 X 2 Y + X Y 2 + 5 Y C.

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 basic algebraic operations. We will also examine a specific problem and determine the correct answer among the given options.

Understanding Polynomial Expressions

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

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

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

Simplifying Polynomial Expressions

To simplify a polynomial expression, we need to combine like terms, which are terms that have the same variable and exponent. We can do this by adding or subtracting the coefficients of like terms.

Example Problem

Let's consider the following polynomial expression:

$$\left(8x^2y^2 - 9x^2y + 9y\right) - \left(6x^2y - xy^2 + 4y\right)$$

Our goal is to simplify this expression by combining like terms.

Step 1: Distribute the Negative Sign

The first step is to distribute the negative sign to each term inside the second set of parentheses:

$$\left(8x^2y^2 - 9x^2y + 9y\right) - 6x^2y + xy^2 - 4y$$

Step 2: Combine Like Terms

Now, we can combine like terms by adding or subtracting the coefficients of like terms:

$$8x^2y^2 - 9x^2y - 6x^2y + 9y - 4y + xy^2$$

Step 3: Simplify the Expression

We can simplify the expression by combining the like terms:

$$8x^2y^2 - 15x^2y + xy^2 + 5y$$

Conclusion

In conclusion, we have simplified the given polynomial expression by combining like terms and applying basic algebraic operations. The correct answer is:

$8x^2y^2 - 15x^2y + xy^2 + 5y$

This is option B.

Discussion

This problem requires a basic understanding of polynomial expressions and algebraic operations. The student should be able to identify like terms and combine them to simplify the expression. This problem is suitable for students who have a basic understanding of algebra and are looking to improve their skills in simplifying polynomial expressions.

Tips and Variations

• To make this problem more challenging, you can add more terms to the polynomial expression or use different variables.
• To make this problem easier, you can provide a simplified version of the polynomial expression and ask the student to identify the like terms.
• You can also use this problem as a starting point to explore other algebraic operations, such as multiplying and dividing polynomial expressions.

Common Mistakes

• Failing to distribute the negative sign to each term inside the second set of parentheses.
• Failing to combine like terms correctly.
• Not simplifying the expression after combining like terms.

Conclusion

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Q: What is a polynomial expression?

A: A polynomial expression is a mathematical expression consisting of variables and coefficients combined using addition, subtraction, and multiplication.

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

A: A polynomial expression is a specific type of algebraic expression that consists of variables and coefficients combined using addition, subtraction, and multiplication. An algebraic expression, on the other hand, can include any combination of variables, coefficients, and mathematical operations.

Q: How do I simplify a polynomial expression?

A: To simplify a polynomial expression, you need to combine like terms, which are terms that have the same variable and exponent. You can do this by adding or subtracting the coefficients of like terms.

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

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

1. Distribute the negative sign to each term inside the second set of parentheses.
2. Combine like terms by adding or subtracting the coefficients of like terms.
3. Simplify the expression by combining the like terms.

Q: How do I identify like terms in a polynomial expression?

A: To identify like terms in a polynomial expression, you need to look for terms that have the same variable and exponent. For example, in the expression $2x^2 + 3x^2$, the terms $2x^2$ and $3x^2$ are like terms because they have the same variable ($x$) and exponent ($2$).

Q: Can I simplify a polynomial expression by combining unlike terms?

A: No, you cannot simplify a polynomial expression by combining unlike terms. Unlike terms are terms that have different variables or exponents, and combining them would result in an invalid expression.

Q: How do I simplify a polynomial expression with multiple variables?

A: To simplify a polynomial expression with multiple variables, you need to combine like terms by adding or subtracting the coefficients of like terms. For example, in the expression $2x^2y + 3x^2y$, the terms $2x^2y$ and $3x^2y$ are like terms because they have the same variables ($x$ and $y$) and exponent ($2$).

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

A: Yes, you can simplify a polynomial expression with a negative coefficient by distributing the negative sign to each term inside the second set of parentheses and then combining like terms.

Q: How do I check my work when simplifying a polynomial expression?

A: To check your work when simplifying a polynomial expression, you need to:

1. Re-read the original expression to ensure that you have not missed any terms.
2. Verify that you have combined like terms correctly.
3. Simplify the expression by combining the like terms.
4. Check that the simplified expression is valid and makes sense.

Q: What are some common mistakes to avoid when simplifying polynomial expressions?

A: Some common mistakes to avoid when simplifying polynomial expressions include:

• Failing to distribute the negative sign to each term inside the second set of parentheses.
• Failing to combine like terms correctly.
• Not simplifying the expression after combining like terms.
• Combining unlike terms.

Conclusion

In conclusion, simplifying polynomial expressions is a crucial skill for any math enthusiast. By following the steps outlined in this article and avoiding common mistakes, you can simplify polynomial expressions and improve your skills in algebra.