Simplify: \[$\left(2v^2 + 7v + 8\right) - \left(2 - 7v^3 - 6v\right)\$\]A) \[$7v^3 + 2v^2 + 13v + 6\$\]B) \[$v^3 + 6v^2 + 13v + 6\$\]C) \[$v^3 + 2v^2 + 13v + 6\$\]D) \[$7v^3 + 6v^2 + 13v + 6\$\]

Understanding the Problem

The given problem involves simplifying an algebraic expression by combining like terms. To simplify the expression, we need to apply the rules of arithmetic operations, specifically the distributive property and the order of operations.

Step 1: Distribute the Negative Sign

The first step is to distribute the negative sign to the terms inside the second set of parentheses. This will change the signs of all the terms inside the parentheses.

$\left(2v^2 + 7v + 8\right) - \left(2 - 7v^3 - 6v\right) = 2v^2 + 7v + 8 - 2 + 7v^3 + 6v$

Step 2: Combine Like Terms

Now, we need to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have terms with the variable $v$ raised to the power of 3, 2, and 1.

$2v^2 + 7v + 8 - 2 + 7v^3 + 6v = 7v^3 + (2v^2 - 0) + (7v + 6v) + (8 - 2)$

Step 3: Simplify the Expression

Now, we can simplify the expression by combining the like terms.

$7v^3 + (2v^2 - 0) + (7v + 6v) + (8 - 2) = 7v^3 + 2v^2 + 13v + 6$

Conclusion

The simplified expression is $7v^3 + 2v^2 + 13v + 6$. This is the correct answer.

Answer

The correct answer is:

• A) $7v^3 + 2v^2 + 13v + 6$

Explanation

The correct answer is obtained by simplifying the given expression using the rules of arithmetic operations. The expression is simplified by distributing the negative sign, combining like terms, and simplifying the resulting expression.

Tips and Tricks

• When simplifying algebraic expressions, it is essential to apply the rules of arithmetic operations in the correct order.
• Distributing the negative sign is a crucial step in simplifying expressions with multiple terms.
• Combining like terms is a key step in simplifying expressions and should be done carefully to avoid errors.

Practice Problems

• Simplify the expression: $\left(3x^2 + 2x - 1\right) - \left(2x^2 - 3x + 4\right)$
• Simplify the expression: $\left(2y^3 + 5y^2 - 3\right) - \left(y^3 - 2y^2 + 1\right)$

Solutions

• The simplified expression is $x^2 + 5x - 5$.
• The simplified expression is $y^3 + 7y^2 - 4$.

Conclusion

Q&A: Simplifying Algebraic Expressions

Q: What is the first step in simplifying an algebraic expression? A: The first step in simplifying an algebraic expression is to distribute the negative sign to the terms inside the second set of parentheses.

Q: What is the distributive property? A: The distributive property is a rule of arithmetic operations that states that a single term can be distributed to multiple terms inside parentheses.

Q: How do I combine like terms? A: To combine like terms, we need to identify the terms with the same variable raised to the same power and add or subtract their coefficients.

Q: What is the order of operations? A: The order of operations is a set of rules that dictates the order in which we perform arithmetic operations in an expression. The order of operations is:

1. Parentheses
2. Exponents
3. Multiplication and Division
4. Addition and Subtraction

Q: How do I simplify an expression with multiple terms? A: To simplify an expression with multiple terms, we need to apply the distributive property, combine like terms, and simplify the resulting expression.

Q: What is the difference between a variable and a constant? A: A variable is a symbol that represents a value that can change, while a constant is a value that does not change.

Q: How do I simplify an expression with variables and constants? A: To simplify an expression with variables and constants, we need to apply the distributive property, combine like terms, and simplify the resulting expression.

Q: What is the final answer to the original problem? A: The final answer to the original problem is $7v^3 + 2v^2 + 13v + 6$.

Q: What are some common mistakes to avoid when simplifying algebraic expressions? A: Some common mistakes to avoid when simplifying algebraic expressions include:

• Not distributing the negative sign correctly
• Not combining like terms correctly
• Not simplifying the resulting expression correctly

Q: How can I practice simplifying algebraic expressions? A: You can practice simplifying algebraic expressions by working through practice problems and checking your answers with a calculator or a teacher.

Q: What are some real-world applications of simplifying algebraic expressions? A: Simplifying algebraic expressions has many real-world applications, including:

• Solving systems of equations
• Finding the maximum or minimum value of a function
• Modeling real-world phenomena

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics. By applying the rules of arithmetic operations and combining like terms, we can simplify complex expressions and arrive at the correct answer. Practice problems and solutions are provided to help reinforce the concepts learned in this article.