Simplifying Expressions with Variables

Oct 13, 2025 by magicalceylontours.com 99 views

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Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities. One of the most common techniques used to simplify expressions is combining like terms. In this article, we will focus on simplifying the expression $\left(x^3 - 2x^2 + 2x + 5\right) + (x + 1) =$ . We will break down the process into manageable steps and provide a clear explanation of each step.

Understanding Like Terms

Before we start simplifying the expression, it's essential to understand what like terms are. Like terms are terms that have the same variable raised to the same power. For example, $x^2$ and $2x^2$ are like terms because they both have the variable $x$ raised to the power of 2. Similarly, $3x$ and $-2x$ are like terms because they both have the variable $x$ raised to the power of 1.

Step 1: Distribute the Negative Sign

The first step in simplifying the expression is to distribute the negative sign to the terms inside the parentheses. This will change the sign of each term inside the parentheses.

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

Step 2: Combine Like Terms

Now that we have distributed the negative sign, we can combine like terms. We will start by combining the terms with the same variable raised to the same power.

$x^3$ is a term with no like terms, so we will leave it as is.

$-2x^2$ is a term with no like terms, so we will leave it as is.

$2x$ and $x$ are like terms because they both have the variable $x$ raised to the power of 1. We will combine them by adding their coefficients.

$2x + x = 3x$

$5$ is a constant term with no like terms, so we will leave it as is.

$1$ is a constant term with no like terms, so we will leave it as is.

Step 3: Simplify the Expression

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

$x^3 - 2x^2 + 3x + 5 + 1$

Step 4: Combine Constant Terms

The final step in simplifying the expression is to combine the constant terms.

$5 + 1 = 6$

So, the simplified expression is:

$x^3 - 2x^2 + 3x + 6$

Conclusion

Simplifying expressions is an essential skill in algebra that helps us solve equations and inequalities. By combining like terms, we can simplify complex expressions and make them easier to work with. In this article, we simplified the expression $\left(x^3 - 2x^2 + 2x + 5\right) + (x + 1) =$ by distributing the negative sign, combining like terms, and simplifying the expression.

Frequently Asked Questions

What are like terms? Like terms are terms that have the same variable raised to the same power.
How do I combine like terms? To combine like terms, add the coefficients of the terms with the same variable raised to the same power.
What is the final simplified expression? The final simplified expression is $x^3 - 2x^2 + 3x + 6$ .

Introduction

In our previous article, we simplified the expression $\left(x^3 - 2x^2 + 2x + 5\right) + (x + 1) =$ . We broke down the process into manageable steps and provided a clear explanation of each step. In this article, we will answer some of the most frequently asked questions about simplifying expressions and combining like terms.

Q&A

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, $x^2$ and $2x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

Q: How do I combine like terms?

A: To combine like terms, add the coefficients of the terms with the same variable raised to the same power. For example, $2x$ and $x$ are like terms because they both have the variable $x$ raised to the power of 1. We can combine them by adding their coefficients: $2x + x = 3x$ .

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable raised to the same power. Unlike terms are terms that have different variables or different powers of the same variable. For example, $x$ and $y$ are unlike terms because they have different variables.

Q: Can I combine like terms with fractions?

A: Yes, you can combine like terms with fractions. For example, $\frac{2x}{3}$ and $\frac{x}{3}$ are like terms because they both have the variable $x$ raised to the power of 1. We can combine them by adding their numerators: $\frac{2x}{3} + \frac{x}{3} = \frac{3x}{3} = x$ .

Q: Can I combine like terms with exponents?

A: Yes, you can combine like terms with exponents. For example, $x^2$ and $2x^2$ are like terms because they both have the variable $x$ raised to the power of 2. We can combine them by adding their coefficients: $x^2 + 2x^2 = 3x^2$ .

Q: What is the final simplified expression?

A: The final simplified expression is $x^3 - 2x^2 + 3x + 6$ .

Q: How do I know if I have combined like terms correctly?

A: To check if you have combined like terms correctly, make sure that you have added the coefficients of the terms with the same variable raised to the same power. You can also use the distributive property to check your work.

Q: Can I simplify expressions with variables and constants?

A: Yes, you can simplify expressions with variables and constants. For example, $x + 2$ is an expression with a variable and a constant. We can simplify it by combining the variable and the constant: $x + 2 = x + 2$ .

Conclusion

Simplifying expressions is an essential skill in algebra that helps us solve equations and inequalities. By combining like terms, we can simplify complex expressions and make them easier to work with. In this article, we answered some of the most frequently asked questions about simplifying expressions and combining like terms.

Frequently Asked Questions

What are like terms? Like terms are terms that have the same variable raised to the same power.
How do I combine like terms? To combine like terms, add the coefficients of the terms with the same variable raised to the same power.
What is the difference between like terms and unlike terms? Like terms are terms that have the same variable raised to the same power. Unlike terms are terms that have different variables or different powers of the same variable.
Can I combine like terms with fractions? Yes, you can combine like terms with fractions.
Can I combine like terms with exponents? Yes, you can combine like terms with exponents.