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

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 consisting of variables and coefficients combined using addition, subtraction, and multiplication. The variables in a polynomial expression can be raised to various powers, and the coefficients can be any real number.

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 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:

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

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

Step 1: Distribute the $v^2$ Term

The first step is to distribute the $v^2$ term to the terms inside the parentheses:

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

Step 2: Combine Like Terms

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

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

Step 3: Simplify the Expression

We can simplify the expression by combining like terms:

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

Step 4: Final Simplification

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

$14 v^4 + v^4 w^2 - 11 v^2 w^3$

However, we notice that the original expression has a $-2 v^2$ term, which we missed in our simplification. To correct this, we need to add $2 v^2$ to the expression:

$14 v^4 + v^4 w^2 - 11 v^2 w^3 + 2$

Conclusion

In conclusion, the simplified expression is:

$14 v^4 + v^4 w^2 - 11 v^2 w^3 + 2$

This expression is equivalent to the given polynomial expression.

Answer

The correct answer is:

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

However, we simplified the expression to:

$14 v^4 + v^4 w^2 - 11 v^2 w^3 + 2$

Which is not the same as the answer choice A. We made an error in our simplification.

Discussion

The correct answer is actually:

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

However, we simplified the expression to:

$14 v^4 + v^4 w^2 - 11 v^2 w^3 + 2$

Which is not the same as the answer choice A. We made an error in our simplification.

Correcting the Error

To correct the error, we need to re-examine the expression and simplify it correctly. Let's re-simplify the expression:

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

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

$= 14 v^4 + v^4 w^2 + 15 v^2 w^3 + 2$

Conclusion

In conclusion, the correct answer is:

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

Introduction

In our previous article, we explored how to simplify polynomial expressions by combining like terms and applying the distributive property. We also examined a specific example and determined which expression is equivalent to the given polynomial expression. In this article, we will provide a Q&A guide to help you better understand the concept of simplifying polynomial expressions.

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. The variables in a polynomial expression can be raised to various powers, and the coefficients can be any real number.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that the product of a single term and a sum of terms is equal to the sum of the products of the single term and each of the terms in the sum.

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 raised to the same power. You can combine like terms by adding or subtracting their coefficients.

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, is a more general term that includes polynomial expressions, rational expressions, and other types of expressions.

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

A: Yes, you can simplify a polynomial expression with negative coefficients. When simplifying a polynomial expression with negative coefficients, you need to remember to change the sign of the coefficient when combining like terms.

Q: How do I know when to apply the distributive property?

A: You should apply the distributive property when you have a single term multiplied by a sum of terms. The distributive property allows you to multiply each term in the sum by the single term, making it easier to simplify the expression.

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. When simplifying a polynomial expression with variables raised to different powers, you need to combine like terms and apply the distributive property.

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

A: The final step in simplifying a polynomial expression is to check your work and make sure that you have combined all like terms and applied the distributive property correctly.

Conclusion

In conclusion, simplifying polynomial expressions is a crucial skill for any math enthusiast. By understanding the concept of like terms, the distributive property, and how to apply them, you can simplify polynomial expressions with ease. Remember to always check your work and make sure that you have combined all like terms and applied the distributive property correctly.

Common Mistakes to Avoid

• Not combining like terms correctly
• Not applying the distributive property correctly
• Not checking work for errors
• Not simplifying expressions with negative coefficients correctly

Tips and Tricks

• Use the distributive property to simplify expressions with multiple terms
• Combine like terms by adding or subtracting their coefficients
• Check your work by plugging in values or using a calculator
• Use algebraic properties, such as the commutative and associative properties, to simplify expressions

Practice Problems

• Simplify the following polynomial expression: $\left(2x^2+3\right)+x\left(x^2+2x-3\right)-\left(4x^2+2\right)$
• Simplify the following polynomial expression: $\left(5y^3+2\right)+y^2\left(y^2+2y-3\right)-\left(3y^3+2\right)$
• Simplify the following polynomial expression: $\left(2z^2+3\right)+z\left(z^2+2z-3\right)-\left(4z^2+2\right)$

Answer Key

• $\left(2x^2+3\right)+x\left(x^2+2x-3\right)-\left(4x^2+2\right) = 2x^2+3+x^3+2x-3x^2-4x^2-2 = x^3-5x^2+2x+1$
• $\left(5y^3+2\right)+y^2\left(y^2+2y-3\right)-\left(3y^3+2\right) = 5y^3+2+y^4+2y^3-3y^2-3y^3-2 = y^4+2y^3-3y^2-2$
• $\left(2z^2+3\right)+z\left(z^2+2z-3\right)-\left(4z^2+2\right) = 2z^2+3+z^3+2z-3z^2-4z^2-2 = z^3-5z^2+2z+1$