Perform The Indicated Operation By Removing The Parentheses And Combining Like Terms: \left(x^2+9\right)-\left(-7x^2-3x+9\right ]

Mar 10, 2025 by ADMIN 130 views

**Simplifying Algebraic Expressions: A Step-by-Step Guide**

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 by removing parentheses and combining like terms. We will use the given expression $\left(x^2+9\right)-\left(-7x^2-3x+9\right)$ as an example to demonstrate the steps involved.

Understanding the Expression

Before we begin simplifying the expression, let's take a closer look at what it represents. The expression consists of two parts: $\left(x^2+9\right)$ and $\left(-7x^2-3x+9\right)$ . The first part is a quadratic expression with a positive coefficient, while the second part is a quadratic expression with a negative coefficient.

Removing Parentheses

The first step in simplifying the expression is to remove the parentheses. To do this, we need to distribute the negative sign to each term inside the second set of parentheses. This will give us:

x^2+9+7x^2+3x-9

Combining Like Terms

Now that we have removed the parentheses, we can combine like terms. Like terms are terms that have the same variable and exponent. In this case, we have two like terms: $x^2$ and $7x^2$ . We can combine these terms by adding their coefficients:

x^2+7x^2=8x^2

We also have two like terms: $3x$ and $-9$ . However, $-9$ is a constant term, so we cannot combine it with $3x$ . Therefore, we will leave it as is.

Simplifying the Expression

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

8x^2+3x-9

Conclusion

In this article, we have demonstrated the process of simplifying algebraic expressions by removing parentheses and combining like terms. We used the given expression $\left(x^2+9\right)-\left(-7x^2-3x+9\right)$ as an example to illustrate the steps involved. By following these steps, we can simplify complex algebraic expressions and make them easier to work with.

Tips and Tricks

Here are some tips and tricks to help you simplify algebraic expressions:

Distribute the negative sign: When removing parentheses, make sure to distribute the negative sign to each term inside the parentheses.
Combine like terms: Combine like terms by adding their coefficients.
Leave constants alone: Don't combine constants with variables.
Simplify the expression: Combine the remaining terms to simplify the expression.

Common Mistakes to Avoid

Here are some common mistakes to avoid when simplifying algebraic expressions:

Forgetting to distribute the negative sign: Make sure to distribute the negative sign to each term inside the parentheses.
Not combining like terms: Combine like terms by adding their coefficients.
Combining constants with variables: Don't combine constants with variables.
Not simplifying the expression: Combine the remaining terms to simplify the expression.

Real-World Applications

Simplifying algebraic expressions has many real-world applications. Here are a few examples:

Science and engineering: Algebraic expressions are used to model real-world phenomena, such as the motion of objects and the behavior of electrical circuits.
Economics: Algebraic expressions are used to model economic systems and make predictions about future trends.
Computer science: Algebraic expressions are used to write algorithms and solve problems in computer science.

Conclusion

Introduction

In our previous article, we explored the process of simplifying algebraic expressions by removing parentheses and combining like terms. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q: What is the difference between a like term and a unlike term?

A: A like term is a term that has the same variable and exponent. For example, $x^2$ and $7x^2$ are like terms because they both have the variable $x$ and the exponent $2$ . A unlike term is a term that has a different variable or exponent. For example, $x^2$ and $3y^3$ are unlike terms because they have different variables and exponents.

Q: How do I know which terms to combine?

A: To combine like terms, you need to identify the terms that have the same variable and exponent. Then, you can add their coefficients to combine the terms. For example, in the expression $x^2+7x^2$ , you can combine the terms by adding their coefficients: $x^2+7x^2=8x^2$ .

Q: What happens if I have a negative coefficient?

A: If you have a negative coefficient, you need to distribute the negative sign to each term inside the parentheses. For example, in the expression $-(x^2+7x^2)$ , you need to distribute the negative sign to each term: $-x^2-7x^2$ .

Q: Can I combine constants with variables?

A: No, you cannot combine constants with variables. Constants are numbers that do not have a variable, while variables are letters or symbols that represent a value. For example, in the expression $x^2+3$ , you cannot combine the constant $3$ with the variable $x^2$ .

Q: How do I simplify an expression with multiple sets of parentheses?

A: To simplify an expression with multiple sets of parentheses, you need to follow the order of operations (PEMDAS):

Evaluate any expressions inside the parentheses.
Remove any parentheses that are not necessary.
Combine like terms.
Simplify the expression.

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

A: Yes, you can use a calculator to simplify algebraic expressions. However, it's always a good idea to check your work by hand to make sure you understand the process.

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

A: Some common mistakes to avoid when simplifying algebraic expressions include:

Forgetting to distribute the negative sign.
Not combining like terms.
Combining constants with variables.
Not simplifying the expression.

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

A: An expression is simplified when there are no like terms that can be combined. You can check if an expression is simplified by looking for any like terms that can be combined.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for any math enthusiast. By following the steps outlined in this article, you can simplify complex algebraic expressions and make them easier to work with. Remember to distribute the negative sign, combine like terms, leave constants alone, and simplify the expression. With practice and patience, you will become proficient in simplifying algebraic expressions and be able to apply them to real-world problems.

Additional Resources

If you're looking for additional resources to help you simplify algebraic expressions, here are a few suggestions:

Math textbooks: Math textbooks often have examples and exercises that can help you practice simplifying algebraic expressions.
Online resources: There are many online resources available that can help you simplify algebraic expressions, including video tutorials and interactive exercises.
Math software: Math software such as Mathematica and Maple can help you simplify algebraic expressions and perform other mathematical tasks.

Final Tips

Here are a few final tips to help you simplify algebraic expressions:

Practice, practice, practice: The more you practice simplifying algebraic expressions, the more comfortable you will become with the process.
Use a calculator: A calculator can be a useful tool when simplifying algebraic expressions, but make sure to check your work by hand to make sure you understand the process.
Check your work: Always check your work to make sure you have simplified the expression correctly.