Simplify The Expression:$\[ \begin{array}{l} \left(6x^2 - 2x + 4\right) \\ -\left(3x^2 + X - 1\right) \end{array} \\]

Introduction

In this article, we will simplify the given expression by combining like terms and applying the rules of algebra. The expression to be simplified is:

{ \begin{array}{l} \left(6x^2 - 2x + 4\right) \\ -\left(3x^2 + x - 1\right) \end{array} \}

Understanding the Expression

The given expression consists of two polynomials, each enclosed in parentheses. The first polynomial is $6x^2 - 2x + 4$, and the second polynomial is $3x^2 + x - 1$. To simplify the expression, we need to combine like terms, which means adding or subtracting the coefficients of the same variables.

Step 1: Distribute the Negative Sign

The first step in simplifying the expression is to distribute the negative sign to the second polynomial. This means multiplying each term in the second polynomial by -1.

{ \begin{array}{l} \left(6x^2 - 2x + 4\right) \\ -3x^2 - x + 1 \end{array} \}

Step 2: Combine Like Terms

Now that we have distributed the negative sign, we can combine like terms. The like terms in this expression are the terms with the same variable and exponent. In this case, the like terms are $6x^2$ and $-3x^2$, $-2x$ and $-x$, and $4$ and $1$.

{ \begin{array}{l} \left(6x^2 - 2x + 4\right) \\ -3x^2 - x + 1 \end{array} \}

Combining the like terms, we get:

{ \begin{array}{l} \left(6x^2 - 3x^2 - 2x - x + 4 + 1\right) \end{array} \}

Step 3: Simplify the Expression

Now that we have combined the like terms, we can simplify the expression by combining the coefficients of the same variables.

{ \begin{array}{l} \left(3x^2 - 3x + 5\right) \end{array} \}

Conclusion

In this article, we simplified the given expression by combining like terms and applying the rules of algebra. The simplified expression is $\left(3x^2 - 3x + 5\right)$. This expression cannot be simplified further, as there are no more like terms to combine.

Final Answer

The final answer is $\boxed{\left(3x^2 - 3x + 5\right)}$.

Tips and Tricks

• When simplifying expressions, always start by distributing the negative sign to the second polynomial.
• Combine like terms by adding or subtracting the coefficients of the same variables.
• Simplify the expression by combining the coefficients of the same variables.

Common Mistakes

• Failing to distribute the negative sign to the second polynomial.
• Failing to combine like terms.
• Simplifying the expression incorrectly.

Real-World Applications

Simplifying expressions is an important skill in mathematics, as it allows us to solve equations and inequalities more easily. In real-world applications, simplifying expressions is used in a variety of fields, including physics, engineering, and economics.

Example Problems

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

Solutions

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

{ \begin{array}{l} \left(2x^2 - x^2 + 3x - 2x - 1 - 3\right) \end{array} \}

{ \begin{array}{l} \left(x^2 + x - 4\right) \end{array} \}

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

{ \begin{array}{l} \left(4x^2 + 2x^2 - 2x + 3x + 1 - 2\right) \end{array} \}

{ \begin{array}{l} \left(6x^2 + x - 1\right) \end{array} \}

Conclusion

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

A: The first step in simplifying an expression is to distribute the negative sign to the second polynomial. This means multiplying each term in the second polynomial by -1.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the same variables. For example, if you have two terms with the same variable and exponent, you can combine them by adding or subtracting their coefficients.

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

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

• Failing to distribute the negative sign to the second polynomial
• Failing to combine like terms
• Simplifying the expression incorrectly

Q: How do I know if an expression can be simplified further?

A: To determine if an expression can be simplified further, you need to check if there are any like terms that can be combined. If there are no like terms, then the expression cannot be simplified further.

Q: Can you provide an example of how to simplify an expression?

A: Let's simplify the expression: $\left(2x^2 + 3x - 1\right) - \left(x^2 + 2x + 3\right)$

{ \begin{array}{l} \left(2x^2 - x^2 + 3x - 2x - 1 - 3\right) \end{array} \}

{ \begin{array}{l} \left(x^2 + x - 4\right) \end{array} \}

Q: How do I apply the rules of algebra when simplifying expressions?

A: When simplifying expressions, you need to apply the rules of algebra, which include:

• Distributing the negative sign to the second polynomial
• Combining like terms
• Simplifying the expression by combining the coefficients of the same variables

Q: Can you provide some tips for simplifying expressions?

A: Here are some tips for simplifying expressions:

• Start by distributing the negative sign to the second polynomial
• Combine like terms by adding or subtracting the coefficients of the same variables
• Simplify the expression by combining the coefficients of the same variables
• Check your work to make sure you have not made any mistakes

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

A: To determine if you have simplified an expression correctly, you need to check your work to make sure you have not made any mistakes. You can do this by:

• Checking your calculations to make sure you have not made any errors
• Verifying that you have combined all like terms
• Simplifying the expression again to make sure you have not missed any like terms

Q: Can you provide some real-world applications of simplifying expressions?

A: Simplifying expressions is an important skill in mathematics, as it allows us to solve equations and inequalities more easily. In real-world applications, simplifying expressions is used in a variety of fields, including:

• Physics: Simplifying expressions is used to solve equations of motion and to calculate the energy of a system.
• Engineering: Simplifying expressions is used to design and optimize systems, such as bridges and buildings.
• Economics: Simplifying expressions is used to model economic systems and to make predictions about the behavior of markets.

Q: Can you provide some example problems for simplifying expressions?

A: Here are some example problems for simplifying expressions:

• Simplify the expression: $\left(4x^2 - 2x + 1\right) + \left(2x^2 + 3x - 2\right)$
• Simplify the expression: $\left(2x^2 + 3x - 1\right) - \left(x^2 + 2x + 3\right)$
• Simplify the expression: $\left(6x^2 - 3x + 4\right) + \left(2x^2 - 2x - 1\right)$

Q: Can you provide some solutions to the example problems?

A: Here are the solutions to the example problems:

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

{ \begin{array}{l} \left(4x^2 + 2x^2 - 2x + 3x + 1 - 2\right) \end{array} \}

{ \begin{array}{l} \left(6x^2 + x - 1\right) \end{array} \}

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

{ \begin{array}{l} \left(2x^2 - x^2 + 3x - 2x - 1 - 3\right) \end{array} \}

{ \begin{array}{l} \left(x^2 + x - 4\right) \end{array} \}

• Simplify the expression: $\left(6x^2 - 3x + 4\right) + \left(2x^2 - 2x - 1\right)$

{ \begin{array}{l} \left(6x^2 + 2x^2 - 3x - 2x + 4 - 1\right) \end{array} \}

{ \begin{array}{l} \left(8x^2 - 5x + 3\right) \end{array} \}