Which Expression Is Equivalent To The Given Polynomial Expression? \left(9v^4+2\right) + V^2\left(v^2w^2 + 2w^3 - 2v^2\right) - \left(-13v^2w^3 + 7v^4\right ]A. 16 U 6 W 5 + 2 16u^6w^5 + 2 16 U 6 W 5 + 2 B. 14 V 4 + 16 V 6 W 5 + 2 14v^4 + 16v^6w^5 + 2 14 V 4 + 16 V 6 W 5 + 2 C. $v 4w 2 +

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 the process of simplifying a given polynomial expression and determine which expression is equivalent to the given polynomial expression.

Understanding Polynomial Expressions

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

The Given Polynomial Expression

The given polynomial expression is:

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

Step 1: Distribute the $v^2$ Term

To simplify the given polynomial expression, we need to start by distributing the $v^2$ term inside the parentheses.

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

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

Step 2: Combine Like Terms

Next, we need to combine like terms in the expression.

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

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

$= \left(0v^4\right) + v^4w^2 + 15v^2w^3 + 2$

Step 3: Simplify the Expression

Now, we can simplify the expression by removing any terms that are equal to zero.

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

Step 4: Factor Out Common Terms

Finally, we can factor out common terms from the expression.

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

$= v^2\left(v^2w^2\right) + 15v^2\left(v^2w^3\right) + 2$

$= v^2\left(v^2w^2 + 15v^2w^3\right) + 2$

Conclusion

In conclusion, the simplified expression is:

$v^2\left(v^2w^2 + 15v^2w^3\right) + 2$

This expression is equivalent to the given polynomial expression.

Answer

The correct answer is:

C. $v^4w^2 + 15v^2w^3 + 2$

Discussion

The given polynomial expression can be simplified by distributing the $v^2$ term, combining like terms, and factoring out common terms. The simplified expression is equivalent to the given polynomial expression.

Final Answer

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

Q: How do I simplify a polynomial expression?

A: To simplify a polynomial expression, you need to follow these steps:

1. Distribute any terms that are being multiplied by a variable or a constant.
2. Combine like terms by adding or subtracting terms with the same variable and exponent.
3. Factor out common terms from the expression.
4. Simplify the expression by removing any terms that are equal to zero.

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 only addition, subtraction, and multiplication. An algebraic expression, on the other hand, can include any combination of variables, coefficients, and mathematical operations.

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

A: Yes, you can simplify a polynomial expression with negative exponents. To do this, you need to rewrite the expression with positive exponents by using the rule that $a^{-n} = \frac{1}{a^n}$.

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

A: To simplify a polynomial expression with multiple variables, you need to follow the same steps as before:

1. Distribute any terms that are being multiplied by a variable or a constant.
2. Combine like terms by adding or subtracting terms with the same variable and exponent.
3. Factor out common terms from the expression.
4. Simplify the expression by removing any terms that are equal to zero.

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

A: Yes, you can simplify a polynomial expression with fractional exponents. To do this, you need to rewrite the expression with integer exponents by using the rule that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.

Q: How do I check if two polynomial expressions are equivalent?

A: To check if two polynomial expressions are equivalent, you need to simplify both expressions and compare the results. If the simplified expressions are the same, then the original expressions are equivalent.

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

A: Yes, you can simplify a polynomial expression with absolute value. To do this, you need to rewrite the expression without absolute value by using the rule that $|a| = a$ if $a \geq 0$ and $|a| = -a$ if $a < 0$.

Conclusion

In conclusion, simplifying polynomial expressions is an important skill in algebra that can be used to solve a wide range of problems. By following the steps outlined in this article, you can simplify polynomial expressions with variables, coefficients, and mathematical operations. Remember to always check your work by simplifying both expressions and comparing the results.

Final Answer

The final answer is that simplifying polynomial expressions is a crucial skill in algebra that can be used to solve a wide range of problems. By following the steps outlined in this article, you can simplify polynomial expressions with variables, coefficients, and mathematical operations.