Factor The Expression Completely. { -30 - 21x$}$

Mar 12, 2025 by ADMIN 49 views

**Factoring the Expression Completely: A Step-by-Step Guide**

Introduction

Factoring an expression is a fundamental concept in algebra that involves expressing a given expression as a product of simpler expressions. In this article, we will focus on factoring the expression completely, which means expressing the given expression as a product of prime factors. We will use the expression ${-30 - 21x}$ as an example to demonstrate the step-by-step process of factoring an expression completely.

Understanding the Expression

Before we begin factoring the expression, it is essential to understand the structure of the given expression. The expression ${-30 - 21x}$ consists of two terms: a constant term ${-30}$ and a variable term ${-21x}$ . The constant term is a numerical value, while the variable term is a product of a numerical coefficient and a variable.

Step 1: Factor Out the Greatest Common Factor (GCF)

The first step in factoring the expression is to identify the greatest common factor (GCF) of the two terms. The GCF is the largest numerical value that divides both terms evenly. In this case, the GCF of ${-30}$ and ${-21x}$ is ${-3}$ . To factor out the GCF, we divide each term by the GCF and multiply the result by the GCF.

-30 = -3 × 10
-21x = -3 × 7x

Step 2: Factor the Expression Completely

Now that we have factored out the GCF, we can factor the expression completely. To do this, we need to identify the prime factors of the remaining terms. The prime factors of ${10}$ are ${2}$ and ${5}$ , while the prime factors of ${7x}$ are ${7}$ and ${x}$ .

-30 = -3 × 2 × 5
-21x = -3 × 7 × x

Step 3: Write the Factored Form

Now that we have identified the prime factors of each term, we can write the factored form of the expression. To do this, we multiply the prime factors together.

-30 - 21x = -3 × 2 × 5 - 3 × 7 × x

Simplifying the Factored Form

The factored form of the expression can be simplified by combining like terms. In this case, we can combine the two terms that have a common factor of ${-3}$ .

-30 - 21x = -3(2 × 5 + 7x)

Conclusion

Factoring an expression completely involves identifying the greatest common factor (GCF) of the terms, factoring out the GCF, and then identifying the prime factors of the remaining terms. By following these steps, we can write the factored form of the expression and simplify it by combining like terms. In this article, we used the expression ${-30 - 21x}$ as an example to demonstrate the step-by-step process of factoring an expression completely.

Common Mistakes to Avoid

When factoring an expression completely, there are several common mistakes to avoid. These include:

Not identifying the GCF: Failing to identify the GCF of the terms can lead to incorrect factoring.
Not factoring out the GCF: Failing to factor out the GCF can lead to incorrect factoring.
Not identifying the prime factors: Failing to identify the prime factors of the remaining terms can lead to incorrect factoring.

Tips and Tricks

When factoring an expression completely, there are several tips and tricks to keep in mind. These include:

Use the distributive property: The distributive property states that a(b + c) = ab + ac. This property can be used to expand and simplify expressions.
Use the commutative property: The commutative property states that a + b = b + a. This property can be used to rearrange terms in an expression.
Use the associative property: The associative property states that (a + b) + c = a + (b + c). This property can be used to rearrange terms in an expression.

Real-World Applications

Factoring an expression completely has several real-world applications. These include:

Simplifying algebraic expressions: Factoring an expression completely can be used to simplify algebraic expressions and make them easier to work with.
Solving equations: Factoring an expression completely can be used to solve equations and make them easier to work with.
Graphing functions: Factoring an expression completely can be used to graph functions and make them easier to work with.

Conclusion

Introduction

Factoring an expression completely is a fundamental concept in algebra that involves expressing a given expression as a product of simpler expressions. In our previous article, we discussed the step-by-step process of factoring an expression completely using the expression ${-30 - 21x}$ as an example. In this article, we will provide a Q&A guide to help you understand the concept of factoring an expression completely and address any questions or concerns you may have.

Q: What is factoring an expression completely?

A: Factoring an expression completely involves expressing a given expression as a product of simpler expressions. This involves identifying the greatest common factor (GCF) of the terms, factoring out the GCF, and then identifying the prime factors of the remaining terms.

Q: Why is factoring an expression completely important?

A: Factoring an expression completely is important because it allows us to simplify algebraic expressions and make them easier to work with. It also helps us to solve equations and graph functions.

Q: How do I identify the greatest common factor (GCF) of the terms?

A: To identify the GCF of the terms, you need to find the largest numerical value that divides both terms evenly. You can use the following steps to identify the GCF:

List the factors of each term.
Identify the common factors of the two terms.
Choose the largest common factor as the GCF.

Q: How do I factor out the GCF?

A: To factor out the GCF, you need to divide each term by the GCF and multiply the result by the GCF. This will give you the factored form of the expression.

Q: What are the prime factors of an expression?

A: The prime factors of an expression are the prime numbers that multiply together to give the expression. For example, the prime factors of ${10}$ are ${2}$ and ${5}$ .

Q: How do I identify the prime factors of an expression?

A: To identify the prime factors of an expression, you need to find the prime numbers that multiply together to give the expression. You can use the following steps to identify the prime factors:

List the factors of the expression.
Identify the prime factors of the expression.
Write the prime factors as a product of prime numbers.

Q: What is the difference between factoring and simplifying an expression?

A: Factoring an expression involves expressing a given expression as a product of simpler expressions, while simplifying an expression involves combining like terms to make the expression easier to work with.

Q: Can I factor an expression that has a variable in the denominator?

A: No, you cannot factor an expression that has a variable in the denominator. This is because the variable in the denominator will not cancel out when you factor the expression.

Q: Can I factor an expression that has a negative sign in front of it?

A: Yes, you can factor an expression that has a negative sign in front of it. The negative sign will not affect the factoring process.

Q: How do I know if an expression can be factored completely?

A: To determine if an expression can be factored completely, you need to check if the expression can be written as a product of prime factors. If the expression can be written as a product of prime factors, then it can be factored completely.

Conclusion

In conclusion, factoring an expression completely is a fundamental concept in algebra that involves expressing a given expression as a product of simpler expressions. By following the steps outlined in this article, you can factor an expression completely and simplify it by combining like terms. We hope this Q&A guide has helped you understand the concept of factoring an expression completely and address any questions or concerns you may have.