Find The Product And Simplify Your Answer.$-6\left(4 F^2+4 F-3\right$\]\[$\square\$\]

Feb 27, 2025 by ADMIN 86 views

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

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the process to solve various mathematical problems. In this article, we will focus on simplifying the given expression: $-6\left(4 f^2+4 f-3\right)$ . We will break down the process into manageable steps, making it easier to understand and apply.

Understanding the Expression

Before we start simplifying the expression, let's understand what it means. The given expression is a quadratic expression, which is a polynomial of degree two. It consists of three terms: $4 f^2$ , $4 f$ , and $-3$ . The expression is enclosed in parentheses and multiplied by $-6$ .

Step 1: Distribute the Negative 6

To simplify the expression, we need to distribute the negative 6 to each term inside the parentheses. This means we will multiply each term by $-6$ . Let's start by distributing the negative 6 to the first term: $4 f^2$ .

-6 \times 4 f^2 = -24 f^2

Step 2: Distribute the Negative 6 to the Second Term

Next, we will distribute the negative 6 to the second term: $4 f$ .

-6 \times 4 f = -24 f

Step 3: Distribute the Negative 6 to the Third Term

Finally, we will distribute the negative 6 to the third term: $-3$ .

-6 \times -3 = 18

Step 4: Combine Like Terms

Now that we have distributed the negative 6 to each term, we can combine like terms. In this case, we have two terms with the variable $f$ : $-24 f^2$ and $-24 f$ . We can combine these terms by adding their coefficients.

-24 f^2 + (-24 f) = -24 f^2 - 24 f

We also have a constant term: $18$ . Since there are no other like terms, we can leave it as is.

Step 5: Simplify the Expression

Now that we have combined like terms, we can simplify the expression by writing it in its simplest form.

-6\left(4 f^2+4 f-3\right) = -24 f^2 - 24 f + 18

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the process to solve various mathematical problems. In this article, we focused on simplifying the given expression: $-6\left(4 f^2+4 f-3\right)$ . We broke down the process into manageable steps, making it easier to understand and apply. By following these steps, you can simplify any algebraic expression and solve various mathematical problems.

Tips and Tricks

Always start by distributing the negative sign or coefficient to each term inside the parentheses.
Combine like terms by adding their coefficients.
Simplify the expression by writing it in its simplest form.

Common Mistakes to Avoid

Failing to distribute the negative sign or coefficient to each term inside the parentheses.
Not combining like terms.
Not simplifying the expression by writing it in its simplest form.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications. For example, in physics, you may need to simplify expressions to solve problems involving motion, energy, and momentum. In engineering, you may need to simplify expressions to design and optimize systems. In economics, you may need to simplify expressions to model and analyze economic systems.

Final Thoughts

Introduction

In our previous article, we discussed the process of simplifying algebraic expressions. In this article, we will provide a Q&A guide to help you understand and apply the concepts. We will cover common questions and scenarios that may arise when 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 or coefficient to each term inside the parentheses.

Q: How do I distribute the negative sign or coefficient to each term?

A: To distribute the negative sign or coefficient to each term, you need to multiply each term by the negative sign or coefficient. For example, if you have the expression $-6\left(4 f^2+4 f-3\right)$ , you would multiply each term inside the parentheses by $-6$ .

Q: What is the difference between combining like terms and simplifying an expression?

A: Combining like terms involves adding or subtracting terms that have the same variable and exponent. Simplifying an expression involves rewriting it in its simplest form by combining like terms and eliminating any unnecessary terms.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms that have the same variable and exponent. For example, if you have the expression $-24 f^2 - 24 f + 18$ , you can combine the like terms $-24 f^2$ and $-24 f$ by adding their coefficients.

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

A: The final step in simplifying an algebraic expression is to simplify the expression by writing it in its simplest form. This involves combining like terms and eliminating any unnecessary terms.

Q: Can I simplify an expression with multiple variables?

A: Yes, you can simplify an expression with multiple variables. The process is the same as simplifying an expression with a single variable. You need to distribute the negative sign or coefficient to each term, combine like terms, and simplify the expression.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you need to rewrite the expression with a positive exponent. For example, if you have the expression $x^{-2}$ , you can rewrite it as $\frac{1}{x^2}$ .

Q: Can I simplify an expression with a fraction?

A: Yes, you can simplify an expression with a fraction. The process is the same as simplifying an expression with a single variable. You need to distribute the negative sign or coefficient to each term, combine like terms, and simplify the expression.

Q: How do I simplify an expression with a radical?

A: To simplify an expression with a radical, you need to rewrite the expression with a rational exponent. For example, if you have the expression $\sqrt{16}$ , you can rewrite it as $4$ .

Q: Can I simplify an expression with a trigonometric function?

A: Yes, you can simplify an expression with a trigonometric function. The process is the same as simplifying an expression with a single variable. You need to distribute the negative sign or coefficient to each term, combine like terms, and simplify the expression.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the process to solve various mathematical problems. By following the steps outlined in this article, you can simplify any algebraic expression and solve various mathematical problems. Remember to always start by distributing the negative sign or coefficient to each term, combine like terms, and simplify the expression by writing it in its simplest form.