Factor The Expression.$ 12x^5 + 6x^2 - 18 }$Options A. { 12(x^5 + 6x^2 - 18) $ $B. { 6(2x^5 + X^2 - 3) $}$C. { 6(2x^5 + 6x^2 - 18) $}$D. { 6x 2(2x 3 + X - 3) $}$

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 $12x^5 + 6x^2 - 18$. We will explore the different options provided and determine the correct factorization of the given expression.

Understanding the Expression

Before we proceed with factoring the expression, let's take a closer look at the given expression: $12x^5 + 6x^2 - 18$. The expression consists of three terms: $12x^5$, $6x^2$, and $-18$. We can see that the first two terms have a common factor of $6x^2$, while the third term is a constant.

Option A: $12(x^5 + 6x^2 - 18)$

Let's examine the first option: $12(x^5 + 6x^2 - 18)$. We can see that the expression inside the parentheses is $x^5 + 6x^2 - 18$. However, this expression cannot be factored further using the distributive property.

Option B: $6(2x^5 + x^2 - 3)$

The second option is $6(2x^5 + x^2 - 3)$. We can see that the expression inside the parentheses is $2x^5 + x^2 - 3$. However, this expression cannot be factored further using the distributive property.

Option C: $6(2x^5 + 6x^2 - 18)$

The third option is $6(2x^5 + 6x^2 - 18)$. We can see that the expression inside the parentheses is $2x^5 + 6x^2 - 18$. However, this expression cannot be factored further using the distributive property.

Option D: $6x^2(2x^3 + x - 3)$

The fourth option is $6x^2(2x^3 + x - 3)$. We can see that the expression inside the parentheses is $2x^3 + x - 3$. However, this expression can be factored further using the distributive property.

Factoring the Expression

To factor the expression $12x^5 + 6x^2 - 18$, we need to find the greatest common factor (GCF) of the three terms. The GCF of $12x^5$, $6x^2$, and $-18$ is $6$. Therefore, we can factor out $6$ from each term:

$$12x^5 + 6x^2 - 18 = 6(2x^5 + x^2 - 3)$$

However, we can factor the expression further by factoring out $x^2$ from the first two terms:

$$12x^5 + 6x^2 - 18 = 6x^2(2x^3 + x - 3)$$

Therefore, the correct factorization of the expression $12x^5 + 6x^2 - 18$ is $6x^2(2x^3 + x - 3)$.

Conclusion

In conclusion, factoring an expression involves expressing a given expression as a product of simpler expressions. We examined four different options for factoring the expression $12x^5 + 6x^2 - 18$ and determined that the correct factorization is $6x^2(2x^3 + x - 3)$. We hope this article has provided a clear understanding of how to factor expressions and has helped you to improve your algebra skills.

Key Takeaways

• Factoring an expression involves expressing a given expression as a product of simpler expressions.
• The greatest common factor (GCF) of the terms in an expression is the largest expression that divides each term without leaving a remainder.
• To factor an expression, we need to find the GCF of the terms and factor it out.
• Factoring an expression can help us to simplify complex expressions and make them easier to work with.

Common Mistakes to Avoid

• Not finding the greatest common factor (GCF) of the terms in an expression.
• Factoring out the wrong term or expression.
• Not factoring the expression further by factoring out other common factors.

Real-World Applications

Factoring expressions has many real-world applications in fields such as engineering, economics, and computer science. For example, factoring expressions can help us to:

• Simplify complex mathematical models and make them easier to work with.
• Analyze and understand complex systems and relationships.
• Make predictions and forecasts based on data and trends.

Final Thoughts

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Introduction

Factoring expressions is a fundamental concept in algebra that involves expressing a given expression as a product of simpler expressions. In our previous article, we explored the different options for factoring the expression $12x^5 + 6x^2 - 18$ and determined that the correct factorization is $6x^2(2x^3 + x - 3)$. In this article, we will answer some of the most frequently asked questions about factoring expressions.

Q: What is factoring?

A: Factoring is the process of expressing a given expression as a product of simpler expressions. It involves finding the greatest common factor (GCF) of the terms in an expression and factoring it out.

Q: Why is factoring important?

A: Factoring is important because it helps us to simplify complex expressions and make them easier to work with. It also helps us to identify the underlying structure of an expression and make predictions and forecasts based on data and trends.

Q: How do I factor an expression?

A: To factor an expression, you need to follow these steps:

1. Find the greatest common factor (GCF) of the terms in the expression.
2. Factor out the GCF from each term.
3. Simplify the expression by combining like terms.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest expression that divides each term in an expression without leaving a remainder. It is the product of the common factors of the terms in the expression.

Q: How do I find the GCF of an expression?

A: To find the GCF of an expression, you need to follow these steps:

1. List the factors of each term in the expression.
2. Identify the common factors of the terms.
3. Multiply the common factors together to find the GCF.

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

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

• Not finding the greatest common factor (GCF) of the terms in an expression.
• Factoring out the wrong term or expression.
• Not factoring the expression further by factoring out other common factors.

Q: How do I check my work when factoring expressions?

A: To check your work when factoring expressions, you need to follow these steps:

1. Multiply the factors together to see if you get the original expression.
2. Check if the expression is simplified and if like terms are combined.
3. Check if the GCF is correct and if it is factored out correctly.

Q: What are some real-world applications of factoring expressions?

A: Some real-world applications of factoring expressions include:

• Simplifying complex mathematical models and making them easier to work with.
• Analyzing and understanding complex systems and relationships.
• Making predictions and forecasts based on data and trends.

Q: Can I factor expressions with variables?

A: Yes, you can factor expressions with variables. In fact, factoring expressions with variables is a fundamental concept in algebra. You can use the same steps to factor expressions with variables as you would with numerical expressions.

Q: Can I factor expressions with fractions?

A: Yes, you can factor expressions with fractions. In fact, factoring expressions with fractions is a common technique used in algebra. You can use the same steps to factor expressions with fractions as you would with numerical expressions.

Conclusion

In conclusion, factoring expressions is a fundamental concept in algebra that involves expressing a given expression as a product of simpler expressions. We answered some of the most frequently asked questions about factoring expressions and provided some tips and tricks for factoring expressions. We hope this article has provided a clear understanding of how to factor expressions and has helped you to improve your algebra skills.

Key Takeaways

• Factoring is the process of expressing a given expression as a product of simpler expressions.
• The greatest common factor (GCF) is the largest expression that divides each term in an expression without leaving a remainder.
• To factor an expression, you need to find the GCF and factor it out.
• Factoring expressions is a fundamental concept in algebra that has many real-world applications.

Common Mistakes to Avoid

• Not finding the greatest common factor (GCF) of the terms in an expression.
• Factoring out the wrong term or expression.
• Not factoring the expression further by factoring out other common factors.

Real-World Applications

• Simplifying complex mathematical models and making them easier to work with.
• Analyzing and understanding complex systems and relationships.
• Making predictions and forecasts based on data and trends.