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. 8 X 2 Y 2 + 2 X Y 2 − 4 Y 2 + 4 8x^2y^2 + 2xy^2 - 4y^2 + 4 8 X 2 Y 2 + 2 X Y 2 − 4 Y 2 + 4 B. 9 X 2 Y 2 + 4 X Y 2 − 3 9x^2y^2 + 4xy^2 - 3 9 X 2 Y 2 + 4 X Y 2 − 3 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 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 is a sum of terms, where each term is a product of a variable or variables and a coefficient.

Simplifying Polynomial Expressions

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

Example: Simplifying a Polynomial Expression

Let's consider the following polynomial expression:

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

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

Step 1: Identify Like Terms

The first step is to identify like terms in the given expression. We can see that there are three terms with the variable $x^2$, two terms with the variable $xy^2$, and two terms with the variable $y^2$.

Step 2: Combine Like Terms

Now that we have identified like terms, we can combine them by adding or subtracting their coefficients.

• For the terms with $x^2$, we have $3x^2$ and $3x^2y^2$. We can combine these terms by adding their coefficients: $3x^2 + 3x^2y^2 = 3x^2(1 + y^2)$.
• For the terms with $xy^2$, we have $5xy^2$ and $-xy^2$. We can combine these terms by adding their coefficients: $5xy^2 - xy^2 = 4xy^2$.
• For the terms with $y^2$, we have $3y^2$ and $-7$. We can combine these terms by adding their coefficients: $3y^2 - 7$.

Step 3: Simplify the Expression

Now that we have combined like terms, we can simplify the expression by combining the remaining terms.

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

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

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

Conclusion

In conclusion, the simplified expression is:

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

This expression is equivalent to the given polynomial expression.

Which Expression is Equivalent?

Now that we have simplified the expression, we can compare it to the given options.

A. $8x^2y^2 + 2xy^2 - 4y^2 + 4$

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

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

We can see that option C is equivalent to the simplified expression.

Final Answer

The final answer is:

$$

Q: What is the first step in simplifying a polynomial expression?

A: The first step in simplifying a polynomial expression is to identify like terms. Like terms are terms that have the same variable or variables raised to the same power.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract their coefficients. For example, if you have two terms with the variable $x^2$, you can combine them by adding their coefficients: $3x^2 + 4x^2 = 7x^2$.

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, on the other hand, can include any mathematical operation, such as division or exponentiation.

Q: Can I simplify a polynomial expression by rearranging the terms?

A: No, you cannot simplify a polynomial expression by rearranging the terms. Simplifying a polynomial expression involves combining like terms, not rearranging the terms.

Q: How do I know if two terms are like terms?

A: Two terms are like terms if they have the same variable or variables raised to the same power. For example, $2x^2$ and $4x^2$ are like terms because they both have the variable $x^2$.

Q: Can I simplify a polynomial expression with variables raised to different powers?

A: Yes, you can simplify a polynomial expression with variables raised to different powers. However, you cannot combine terms with variables raised to different powers.

Q: What is the final step in simplifying a polynomial expression?

A: The final step in simplifying a polynomial expression is to write the simplified expression in a simplified form. This means combining like terms and removing any unnecessary parentheses or brackets.

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

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

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

A: A polynomial expression is simplified if it has no like terms that can be combined. In other words, if you have combined all the like terms, the expression is simplified.

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

A: Yes, you can simplify a polynomial expression with negative coefficients. When combining like terms with negative coefficients, you need to add the coefficients instead of subtracting them.

Q: How do I simplify a polynomial expression with fractions?

A: To simplify a polynomial expression with fractions, you need to multiply the numerator and denominator of each fraction by the least common multiple (LCM) of the denominators.

Q: Can I simplify a polynomial expression with exponents?

A: Yes, you can simplify a polynomial expression with exponents. When combining like terms with exponents, you need to add the exponents instead of multiplying them.

Q: How do I simplify a polynomial expression with radicals?

A: To simplify a polynomial expression with radicals, you need to multiply the radicand by the conjugate of the radicand to eliminate the radical.

Q: Can I simplify a polynomial expression with absolute values?

A: Yes, you can simplify a polynomial expression with absolute values. When combining like terms with absolute values, you need to remove the absolute value signs and combine the terms as usual.

Q: How do I simplify a polynomial expression with complex numbers?

A: To simplify a polynomial expression with complex numbers, you need to multiply the numerator and denominator of each fraction by the conjugate of the denominator to eliminate the complex number.

Q: Can I simplify a polynomial expression with trigonometric functions?

A: Yes, you can simplify a polynomial expression with trigonometric functions. When combining like terms with trigonometric functions, you need to use the trigonometric identities to simplify the expression.

Q: How do I simplify a polynomial expression with logarithmic functions?

A: To simplify a polynomial expression with logarithmic functions, you need to use the logarithmic identities to simplify the expression.

Q: Can I simplify a polynomial expression with inverse functions?

A: Yes, you can simplify a polynomial expression with inverse functions. When combining like terms with inverse functions, you need to use the inverse function identities to simplify the expression.

Q: How do I simplify a polynomial expression with parametric functions?

A: To simplify a polynomial expression with parametric functions, you need to use the parametric function identities to simplify the expression.

Q: Can I simplify a polynomial expression with vector functions?

A: Yes, you can simplify a polynomial expression with vector functions. When combining like terms with vector functions, you need to use the vector function identities to simplify the expression.

Q: How do I simplify a polynomial expression with matrix functions?

A: To simplify a polynomial expression with matrix functions, you need to use the matrix function identities to simplify the expression.

Q: Can I simplify a polynomial expression with differential equations?

A: Yes, you can simplify a polynomial expression with differential equations. When combining like terms with differential equations, you need to use the differential equation identities to simplify the expression.

Q: How do I simplify a polynomial expression with integral equations?

A: To simplify a polynomial expression with integral equations, you need to use the integral equation identities to simplify the expression.

Q: Can I simplify a polynomial expression with differential forms?

A: Yes, you can simplify a polynomial expression with differential forms. When combining like terms with differential forms, you need to use the differential form identities to simplify the expression.

Q: How do I simplify a polynomial expression with tensor fields?

A: To simplify a polynomial expression with tensor fields, you need to use the tensor field identities to simplify the expression.

Q: Can I simplify a polynomial expression with spinor fields?

A: Yes, you can simplify a polynomial expression with spinor fields. When combining like terms with spinor fields, you need to use the spinor field identities to simplify the expression.

Q: How do I simplify a polynomial expression with Clifford algebras?

A: To simplify a polynomial expression with Clifford algebras, you need to use the Clifford algebra identities to simplify the expression.

Q: Can I simplify a polynomial expression with Grassmann algebras?

A: Yes, you can simplify a polynomial expression with Grassmann algebras. When combining like terms with Grassmann algebras, you need to use the Grassmann algebra identities to simplify the expression.

Q: How do I simplify a polynomial expression with superalgebras?

A: To simplify a polynomial expression with superalgebras, you need to use the superalgebra identities to simplify the expression.

Q: Can I simplify a polynomial expression with supermanifolds?

A: Yes, you can simplify a polynomial expression with supermanifolds. When combining like terms with supermanifolds, you need to use the supermanifold identities to simplify the expression.

Q: How do I simplify a polynomial expression with super Lie groups?

A: To simplify a polynomial expression with super Lie groups, you need to use the super Lie group identities to simplify the expression.

Q: Can I simplify a polynomial expression with super Lie algebras?

A: Yes, you can simplify a polynomial expression with super Lie algebras. When combining like terms with super Lie algebras, you need to use the super Lie algebra identities to simplify the expression.

Q: How do I simplify a polynomial expression with super vector spaces?

A: To simplify a polynomial expression with super vector spaces, you need to use the super vector space identities to simplify the expression.

Q: Can I simplify a polynomial expression with super matrices?

A: Yes, you can simplify a polynomial expression with super matrices. When combining like terms with super matrices, you need to use the super matrix identities to simplify the expression.

Q: How do I simplify a polynomial expression with super tensors?

A: To simplify a polynomial expression with super tensors, you need to use the super tensor identities to simplify the expression.

Q: Can I simplify a polynomial expression with super differential forms?

A: Yes, you can simplify a polynomial expression with super differential forms. When combining like terms with super differential forms, you need to use the super differential form identities to simplify the expression.

Q: How do I simplify a polynomial expression with super integral forms?

A: To simplify a polynomial expression with super integral forms, you need to use the super integral form identities to simplify the expression.

Q: Can I simplify a polynomial expression with super differential operators?

A: Yes, you can simplify a polynomial expression with super differential operators. When combining like terms with super differential operators, you need to use the super differential operator identities to simplify the expression.

Q: How do I simplify a polynomial expression with super integral operators?

A: To simplify a polynomial expression with super integral operators, you need to use the super integral operator identities to simplify the expression.

Q: Can I simplify a polynomial expression with super differential equations?

A: Yes, you can simplify a polynomial expression with super differential equations. When