Find The Difference. \left(2x^3 + 1\right) - \left(-x^3 + 1\right ] =? X 3 + □ \text{=?}x^3 + \square =? X 3 + □

Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities. One of the fundamental operations in algebra is finding the difference between two expressions. In this article, we will explore how to find the difference between two algebraic expressions, specifically the expression $\left(2x^3 + 1\right) - \left(-x^3 + 1\right)$.

Understanding the Problem

The given expression is a difference of two algebraic expressions. To find the difference, we need to subtract the second expression from the first expression. This involves applying the rules of algebraic operations, such as the distributive property and the properties of exponents.

Step 1: Distribute the Negative Sign

The first step in finding the difference is to distribute the negative sign to the terms inside the second expression. This means multiplying each term inside the second expression by -1.

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

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

Step 2: Combine Like Terms

Now that we have distributed the negative sign, we can combine like terms. Like terms are terms that have the same variable and exponent. In this case, we have two terms with the variable $x^3$ and two constant terms.

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

$= (2x^3 + x^3) + (1 - 1)$

Step 3: Simplify the Expression

Now that we have combined like terms, we can simplify the expression. The expression $(2x^3 + x^3)$ can be simplified by combining the coefficients of $x^3$, which is $2 + 1 = 3$. The expression $(1 - 1)$ simplifies to $0$.

$(2x^3 + x^3) + (1 - 1)$

$= 3x^3 + 0$

Step 4: Write the Final Answer

The final answer is $3x^3$. This is the simplified expression after finding the difference between the two given expressions.

Conclusion

Finding the difference between two algebraic expressions involves applying the rules of algebraic operations, such as the distributive property and the properties of exponents. By distributing the negative sign and combining like terms, we can simplify the expression and find the final answer. In this article, we have explored how to find the difference between the expressions $\left(2x^3 + 1\right) - \left(-x^3 + 1\right)$.

Example Problems

Example 1

Find the difference between the expressions $\left(3x^2 + 2\right) - \left(-2x^2 + 1\right)$.

Solution

To find the difference, we need to distribute the negative sign to the terms inside the second expression.

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

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

Now that we have distributed the negative sign, we can combine like terms.

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

$= (3x^2 + 2x^2) + (2 - 1)$

The expression $(3x^2 + 2x^2)$ can be simplified by combining the coefficients of $x^2$, which is $3 + 2 = 5$. The expression $(2 - 1)$ simplifies to $1$.

$(3x^2 + 2x^2) + (2 - 1)$

$= 5x^2 + 1$

The final answer is $5x^2 + 1$.

Example 2

Find the difference between the expressions $\left(4x + 3\right) - \left(-2x + 1\right)$.

Solution

To find the difference, we need to distribute the negative sign to the terms inside the second expression.

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

$= 4x + 3 + 2x - 1$

Now that we have distributed the negative sign, we can combine like terms.

$4x + 3 + 2x - 1$

$= (4x + 2x) + (3 - 1)$

The expression $(4x + 2x)$ can be simplified by combining the coefficients of $x$, which is $4 + 2 = 6$. The expression $(3 - 1)$ simplifies to $2$.

$(4x + 2x) + (3 - 1)$

$= 6x + 2$

The final answer is $6x + 2$.

Tips and Tricks

• When finding the difference between two expressions, make sure to distribute the negative sign to all the terms inside the second expression.
• Combine like terms by adding or subtracting the coefficients of the same variable and exponent.
• Simplify the expression by combining the coefficients of the same variable and exponent.

Conclusion

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Introduction

In our previous article, we explored how to find the difference between two algebraic expressions. In this article, we will answer some frequently asked questions about finding the difference between two expressions.

Q: What is the difference between two expressions?

A: The difference between two expressions is the result of subtracting one expression from another. It involves applying the rules of algebraic operations, such as the distributive property and the properties of exponents.

Q: How do I find the difference between two expressions?

A: To find the difference between two expressions, follow these steps:

1. Distribute the negative sign to all the terms inside the second expression.
2. Combine like terms by adding or subtracting the coefficients of the same variable and exponent.
3. Simplify the expression by combining the coefficients of the same variable and exponent.

Q: What is the distributive property?

A: The distributive property is a rule of algebra that states that for any numbers a, b, and c, the following equation holds:

a(b + c) = ab + ac

This means that we can distribute the multiplication of a to both b and c.

Q: How do I apply the distributive property?

A: To apply the distributive property, simply multiply the term outside the parentheses to each term inside the parentheses.

For example, if we have the expression 2(x + 3), we can apply the distributive property as follows:

2(x + 3) = 2x + 6

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, 2x and 3x are like terms because they both have the variable x and the exponent 1.

Q: How do I combine like terms?

A: To combine like terms, simply add or subtract the coefficients of the same variable and exponent.

For example, if we have the expression 2x + 3x, we can combine like terms as follows:

2x + 3x = 5x

Q: What is the final answer?

A: The final answer is the simplified expression after finding the difference between the two given expressions.

Q: Can I use a calculator to find the difference between two expressions?

A: Yes, you can use a calculator to find the difference between two expressions. However, it's always a good idea to check your work by hand to make sure you get the correct answer.

Q: What are some common mistakes to avoid when finding the difference between two expressions?

A: Some common mistakes to avoid when finding the difference between two expressions include:

• Not distributing the negative sign to all the terms inside the second expression.
• Not combining like terms.
• Not simplifying the expression by combining the coefficients of the same variable and exponent.

Conclusion

Finding the difference between two algebraic expressions involves applying the rules of algebraic operations, such as the distributive property and the properties of exponents. By distributing the negative sign and combining like terms, we can simplify the expression and find the final answer. In this article, we have answered some frequently asked questions about finding the difference between two expressions and provided tips and tricks to help you practice.

Practice Problems

Problem 1

Find the difference between the expressions $\left(3x^2 + 2\right) - \left(-2x^2 + 1\right)$.

Problem 2

Find the difference between the expressions $\left(4x + 3\right) - \left(-2x + 1\right)$.

Problem 3

Find the difference between the expressions $\left(2x^3 + 1\right) - \left(-x^3 + 1\right)$.

Answer Key

Problem 1

The final answer is $5x^2 + 1$.

Problem 2

The final answer is $6x + 2$.

Problem 3

The final answer is $3x^3$.

Tips and Tricks

• When finding the difference between two expressions, make sure to distribute the negative sign to all the terms inside the second expression.
• Combine like terms by adding or subtracting the coefficients of the same variable and exponent.
• Simplify the expression by combining the coefficients of the same variable and exponent.