Find Each Difference:$\[ \left(-6s^2 + 12s - 8\right) - \left(3s^2 + 8s - 6\right) = \\]A. \[$-9s^2 + 4s - 14\$\]B. \[$-9s^2 + 4s - 2\$\]C. \[$-9s^2 + 20s - 14\$\]D. \[$-9s^2 + 20s - 2\$\]

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will focus on finding the difference between two given algebraic expressions, which involves combining like terms and simplifying the resulting expression. We will use a step-by-step approach to solve the problem and provide a clear explanation of each step.

The Problem

The problem requires us to find the difference between two algebraic expressions:

$$\left(-6s^2 + 12s - 8\right) - \left(3s^2 + 8s - 6\right)$$

Step 1: Distribute the Negative Sign

The first step is to distribute the negative sign to the terms inside the second parentheses. This will change the sign of each term:

$$-3s^2 - 8s + 6$$

Step 2: Combine Like Terms

Now, we need to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have two terms with the variable $s^2$ and two terms with the variable $s$. We can combine these terms by adding or subtracting their coefficients:

$$-6s^2 + 12s - 8 - 3s^2 - 8s + 6$$

Combine the $s^2$ terms:

$$-9s^2 + 12s - 8s - 8 + 6$$

Combine the $s$ terms:

$$-9s^2 + 4s - 2$$

Step 3: Simplify the Expression

The final step is to simplify the expression by combining any remaining like terms. In this case, there are no remaining like terms, so the expression is already simplified:

$$-9s^2 + 4s - 2$$

Conclusion

In this article, we have walked through the steps to simplify the difference between two algebraic expressions. We have distributed the negative sign, combined like terms, and simplified the resulting expression. The final answer is:

$$-9s^2 + 4s - 2$$

This is the correct answer, which matches option B.

Key Takeaways

• Distributing the negative sign changes the sign of each term.
• Combining like terms involves adding or subtracting the coefficients of terms with the same variable raised to the same power.
• Simplifying an expression involves combining any remaining like terms.

Practice Problems

Try simplifying the following expressions:

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

Answer Key

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

Additional Resources

For more practice problems and additional resources, check out the following websites:

• Khan Academy: Algebra
• Mathway: Algebra
• IXL: Algebra

Introduction

In our previous article, we walked through the steps to simplify the difference between two algebraic expressions. We distributed the negative sign, combined like terms, and simplified the resulting expression. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

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

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

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, $2x^2$ and $5x^2$ 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, you add or subtract their coefficients. For example, if you have the terms $2x^2$ and $5x^2$, you can combine them by adding their coefficients: $2x^2 + 5x^2 = 7x^2$.

Q: What is the difference between a coefficient and a variable?

A: A coefficient is a number that is multiplied by a variable. For example, in the term $2x$, the coefficient is 2 and the variable is $x$. A variable is a letter or symbol that represents a value.

Q: Can I simplify an expression by combining non-like terms?

A: No, you cannot simplify an expression by combining non-like terms. Non-like terms are terms that have different variables or different powers of the same variable. For example, $2x^2$ and $3y^2$ are non-like terms because they have different variables.

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

A: The final step in simplifying an algebraic expression is to simplify the resulting expression by combining any remaining like terms.

Q: Can I use a calculator to simplify an algebraic expression?

A: Yes, you can use a calculator to simplify an algebraic expression. However, it's always a good idea to check your work by simplifying the expression by hand.

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
• Combining non-like terms
• Not simplifying the resulting expression
• Not checking your work

Conclusion

Simplifying algebraic expressions is an important skill to master in mathematics. By following the steps outlined in this article and avoiding common mistakes, you can simplify expressions with ease. Remember to distribute the negative sign, combine like terms, and simplify the resulting expression. With practice, you will become proficient in simplifying algebraic expressions and be able to tackle more complex problems.

Practice Problems

Try simplifying the following expressions:

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

Answer Key

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

Additional Resources

For more practice problems and additional resources, check out the following websites:

• Khan Academy: Algebra
• Mathway: Algebra
• IXL: Algebra

By following the steps outlined in this article and practicing with the provided examples, you will become proficient in simplifying algebraic expressions and be able to tackle more complex problems.