Choose The Correct Simplification Of \left(7x^3 - 8x - 5\right) + \left(3x^3 + 7x + 1\right ].A. 10 X 3 − X − 4 10x^3 - X - 4 10 X 3 − X − 4 B. 10 X 3 + X + 4 10x^3 + X + 4 10 X 3 + X + 4 C. 4 X 3 − 15 X − 6 4x^3 - 15x - 6 4 X 3 − 15 X − 6 D. 4 X 3 + 15 X + 6 4x^3 + 15x + 6 4 X 3 + 15 X + 6

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying algebraic expressions, focusing on the correct simplification of the given expression $\left(7x^3 - 8x - 5\right) + \left(3x^3 + 7x + 1\right)$. We will break down the steps involved in simplifying this expression and provide a clear explanation of the correct answer.

Understanding Algebraic Expressions

An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations. It can be a simple expression, such as $2x + 3$, or a more complex expression, like $\left(7x^3 - 8x - 5\right) + \left(3x^3 + 7x + 1\right)$. Algebraic expressions can be simplified by combining like terms, which involves adding or subtracting terms with the same variable and exponent.

Simplifying the Given Expression

To simplify the given expression, we need to combine like terms. The expression consists of two parts: $\left(7x^3 - 8x - 5\right)$ and $\left(3x^3 + 7x + 1\right)$. We can start by combining the terms with the same variable and exponent.

Combining Like Terms

The first step in simplifying the expression is to combine the like terms. We can start by combining the terms with the same variable and exponent.

• Combining the $x^3$ terms: $7x^3 + 3x^3 = 10x^3$
• Combining the $x$ terms: $-8x + 7x = -x$
• Combining the constant terms: $-5 + 1 = -4$

Simplifying the Expression

Now that we have combined the like terms, we can simplify the expression by adding the results.

$\left(7x^3 - 8x - 5\right) + \left(3x^3 + 7x + 1\right) = 10x^3 - x - 4$

Conclusion

In conclusion, the correct simplification of the given expression $\left(7x^3 - 8x - 5\right) + \left(3x^3 + 7x + 1\right)$ is $10x^3 - x - 4$. This simplification was achieved by combining like terms, which involved adding or subtracting terms with the same variable and exponent.

Answer Key

The correct answer is:

A. $10x^3 - x - 4$

Discussion

This problem requires the application of algebraic skills, specifically the ability to combine like terms and simplify expressions. The correct answer can be verified by plugging in values for $x$ and evaluating the expression.

Tips and Tricks

• When simplifying algebraic expressions, it's essential to combine like terms first.
• Make sure to add or subtract terms with the same variable and exponent.
• Use parentheses to group terms and make it easier to combine like terms.

Practice Problems

Try simplifying the following expressions:

1. $\left(2x^2 + 3x - 1\right) + \left(x^2 - 2x + 4\right)$
2. $\left(5x^3 - 2x^2 + 3x - 1\right) + \left(2x^3 + 3x^2 - 2x + 4\right)$

Conclusion

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to combine like terms. Like terms are terms that have the same variable and exponent.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms. For example, if you have $2x^2 + 3x^2$, you can combine them by adding the coefficients: $2x^2 + 3x^2 = 5x^2$.

Q: What is the difference between combining like terms and simplifying an expression?

A: Combining like terms is the process of adding or subtracting terms with the same variable and exponent. Simplifying an expression is the process of reducing an expression to its simplest form by combining like terms and eliminating any unnecessary parentheses or brackets.

Q: Can I simplify an expression by rearranging the terms?

A: No, you cannot simplify an expression by rearranging the terms. Simplifying an expression involves combining like terms and eliminating any unnecessary parentheses or brackets. Rearranging the terms does not change the value of the expression.

Q: How do I know if an expression is already simplified?

A: An expression is already simplified if there are no like terms that can be combined. If an expression has no like terms, it is already in its simplest form.

Q: Can I simplify an expression with variables and constants?

A: Yes, you can simplify an expression with variables and constants. When combining like terms, you need to add or subtract the coefficients of the terms, including the constants.

Q: What is the importance of simplifying algebraic expressions?

A: Simplifying algebraic expressions is important because it helps to:

• Reduce the complexity of the expression
• Make it easier to solve equations and inequalities
• Identify patterns and relationships between variables
• Improve the accuracy and efficiency of calculations

Q: Can I use a calculator to simplify algebraic expressions?

A: Yes, you can use a calculator to simplify algebraic expressions. However, it's essential to understand the underlying math and be able to simplify expressions manually. This will help you to:

• Develop problem-solving skills
• Understand the math behind the calculations
• Identify and correct errors

Q: How do I know if I have simplified an expression correctly?

A: To ensure that you have simplified an expression correctly, you can:

• Check your work by plugging in values for the variables
• Use a calculator to verify the result
• Compare your answer with the original expression to ensure that you have not introduced any errors

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By combining like terms and following the steps outlined in this article, you can simplify even the most complex expressions. Remember to use parentheses to group terms and make it easier to combine like terms. With practice, you'll become proficient in simplifying algebraic expressions and be able to tackle even the most challenging problems.