Select The Correct Answer.Which Expression Is Equivalent To This Polynomial Expression?$\left(8 X^2 Y^2 - 9 X^2 Y + 9 Y\right) - \left(6 X^2 Y - X Y^2 + 4 Y\right$\]A. $-7 X^2 Y^2 + X Y^2 + 5 Y$ B. $8 X^2 Y^2 - 3 X^2 Y - X Y^2 +

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 given polynomial expression by combining like terms and applying basic algebraic operations. 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 consisting of variables, coefficients, and constants, combined using addition, subtraction, and multiplication. Polynomial expressions can be written in the form of:

$$a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$$

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, which are terms that have the same variable and exponent. We can do this by adding or subtracting the coefficients of the like terms.

For example, consider the polynomial expression:

$$2x^2 + 3x + 4 - 2x^2 - x - 1$$

To simplify this expression, we can combine the like terms:

$$2x^2 - 2x^2 + 3x - x + 4 - 1$$

$$= 2x^2 - 2x^2 + 2x + 3 - 1$$

$$= 2x + 2$$

Simplifying the Given Polynomial Expression

Now, let's simplify the given polynomial expression:

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

To simplify this expression, we need to combine the like terms. We can start by distributing the negative sign to the terms inside the second set of parentheses:

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

Now, we can combine the like terms:

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

$$= 8 x^2 y^2 - 15 x^2 y + 9 y - 4 y + x y^2$$

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

Comparing the Simplified Expression with the Options

Now, let's compare the simplified expression with the options:

A. $-7 x^2 y^2 + x y^2 + 5 y$

B. $8 x^2 y^2 - 3 x^2 y - x y^2 + 5 y$

C. $8 x^2 y^2 - 15 x^2 y + 5 y + x y^2$

D. $8 x^2 y^2 - 15 x^2 y + 5 y - x y^2$

The simplified expression matches option C.

Conclusion

In this article, we explored how to simplify a polynomial expression by combining like terms and applying basic algebraic operations. We also examined a specific example and determined which expression is equivalent to the given polynomial expression. By following the steps outlined in this article, you can simplify polynomial expressions with ease and become proficient in algebra.

Final Answer

The final answer is:

$$

Q: What is a polynomial expression?

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

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 the like terms.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, in the expression $2x^2 + 3x + 4 - 2x^2 - x - 1$, the terms $2x^2$ and $-2x^2$ are like terms because they have the same variable ($x$) and exponent ($2$).

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, in the expression $2x^2 + 3x + 4 - 2x^2 - x - 1$, you can combine the like terms as follows:

$$2x^2 - 2x^2 + 3x - x + 4 - 1$$

$$= 2x^2 - 2x^2 + 2x + 3 - 1$$

$$= 2x + 2$$

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, coefficients, and constants, combined using addition, subtraction, and multiplication. An algebraic expression, on the other hand, is a more general term that includes polynomial expressions, rational expressions, and other types of expressions.

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

A: Yes, you can simplify a polynomial expression with variables and constants. For example, consider the expression $2x^2 + 3x + 4 - 2x^2 - x - 1$. You can simplify this expression by combining the like terms:

$$2x^2 - 2x^2 + 3x - x + 4 - 1$$

$$= 2x^2 - 2x^2 + 2x + 3 - 1$$

$$= 2x + 2$$

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

A: Yes, you can simplify a polynomial expression with rational coefficients. For example, consider the expression $\frac{2x^2}{3} + \frac{3x}{2} + \frac{4}{5} - \frac{2x^2}{3} - \frac{x}{2} - \frac{1}{5}$. You can simplify this expression by combining the like terms:

$$\frac{2x^2}{3} - \frac{2x^2}{3} + \frac{3x}{2} - \frac{x}{2} + \frac{4}{5} - \frac{1}{5}$$

$$= \frac{2x^2}{3} - \frac{2x^2}{3} + \frac{2x}{2} + \frac{3}{5}$$

$$= \frac{2x}{2} + \frac{3}{5}$$

$$= x + \frac{3}{5}$$

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

A: Yes, you can simplify a polynomial expression with complex coefficients. For example, consider the expression $2x^2 + 3x + 4i - 2x^2 - x - 2i$. You can simplify this expression by combining the like terms:

$$2x^2 - 2x^2 + 3x - x + 4i - 2i$$

$$= 2x^2 - 2x^2 + 2x + 2i$$

$$= 2x + 2i$$

Conclusion

In this article, we have answered some of the most frequently asked questions about simplifying polynomial expressions. We have covered topics such as like terms, combining like terms, and simplifying polynomial expressions with variables, constants, rational coefficients, and complex coefficients. By following the steps outlined in this article, you can simplify polynomial expressions with ease and become proficient in algebra.