Which Expressions Are Completely Factored? Select Each Correct Answer.A. 24 X 6 − 18 X 5 = 6 X 5 ( 4 X − 3 24x^6 - 18x^5 = 6x^5(4x - 3 24 X 6 − 18 X 5 = 6 X 5 ( 4 X − 3 ]B. 12 X 5 + 8 X 3 = 2 X 3 ( 6 X 2 + 4 12x^5 + 8x^3 = 2x^3(6x^2 + 4 12 X 5 + 8 X 3 = 2 X 3 ( 6 X 2 + 4 ]C. 20 X 3 + 12 X 2 = 4 X 2 ( 5 X + 3 20x^3 + 12x^2 = 4x^2(5x + 3 20 X 3 + 12 X 2 = 4 X 2 ( 5 X + 3 ]D. 18 X 4 − 12 X 2 = 6 X 2 ( 3 X 2 − 2 18x^4 - 12x^2 = 6x^2(3x^2 - 2 18 X 4 − 12 X 2 = 6 X 2 ( 3 X 2 − 2 ]

In algebra, factoring is a process of expressing an algebraic expression as a product of simpler expressions. A completely factored expression is one that cannot be further simplified by factoring. In this article, we will examine four algebraic expressions and determine which ones are completely factored.

What is Factoring?

Factoring is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions. It is a process of breaking down a complex expression into simpler components that can be multiplied together to obtain the original expression. Factoring is an essential tool in algebra, as it allows us to simplify complex expressions, solve equations, and analyze functions.

How to Factor Algebraic Expressions

There are several techniques for factoring algebraic expressions, including:

• Greatest Common Factor (GCF): This involves finding the largest factor that divides each term in the expression.
• Difference of Squares: This involves factoring expressions of the form $a^2 - b^2$.
• Sum and Difference: This involves factoring expressions of the form $a^2 + b^2$ and $a^2 - b^2$.
• Grouping: This involves factoring expressions by grouping terms together.

Examining the Expressions

Let's examine each of the four expressions and determine which ones are completely factored.

A. $24x^6 - 18x^5 = 6x^5(4x - 3)$

To determine if this expression is completely factored, we need to check if it can be further simplified by factoring. The expression $6x^5(4x - 3)$ is already factored, as it is a product of two simpler expressions. However, we can further simplify the expression by factoring out a common factor of $6x^5$. This gives us:

$$24x^6 - 18x^5 = 6x^5(4x - 3)$$

This expression is not completely factored, as it can be further simplified by factoring out a common factor of $6x^5$.

B. $12x^5 + 8x^3 = 2x^3(6x^2 + 4)$

To determine if this expression is completely factored, we need to check if it can be further simplified by factoring. The expression $2x^3(6x^2 + 4)$ is already factored, as it is a product of two simpler expressions. However, we can further simplify the expression by factoring out a common factor of $2x^3$. This gives us:

$$12x^5 + 8x^3 = 2x^3(6x^2 + 4)$$

This expression is not completely factored, as it can be further simplified by factoring out a common factor of $2x^3$.

C. $20x^3 + 12x^2 = 4x^2(5x + 3)$

To determine if this expression is completely factored, we need to check if it can be further simplified by factoring. The expression $4x^2(5x + 3)$ is already factored, as it is a product of two simpler expressions. However, we can further simplify the expression by factoring out a common factor of $4x^2$. This gives us:

$$20x^3 + 12x^2 = 4x^2(5x + 3)$$

This expression is not completely factored, as it can be further simplified by factoring out a common factor of $4x^2$.

D. $18x^4 - 12x^2 = 6x^2(3x^2 - 2)$

To determine if this expression is completely factored, we need to check if it can be further simplified by factoring. The expression $6x^2(3x^2 - 2)$ is already factored, as it is a product of two simpler expressions. However, we can further simplify the expression by factoring out a common factor of $6x^2$. This gives us:

$$18x^4 - 12x^2 = 6x^2(3x^2 - 2)$$

This expression is not completely factored, as it can be further simplified by factoring out a common factor of $6x^2$.

Conclusion

In conclusion, none of the four expressions are completely factored. Each expression can be further simplified by factoring out a common factor. Factoring is an essential tool in algebra, as it allows us to simplify complex expressions, solve equations, and analyze functions.

Key Takeaways

• Factoring is a process of expressing an algebraic expression as a product of simpler expressions.
• A completely factored expression is one that cannot be further simplified by factoring.
• There are several techniques for factoring algebraic expressions, including Greatest Common Factor (GCF), Difference of Squares, Sum and Difference, and Grouping.
• Factoring is an essential tool in algebra, as it allows us to simplify complex expressions, solve equations, and analyze functions.

Final Thoughts

In our previous article, we explored the concept of factoring algebraic expressions and examined four expressions to determine which ones are completely factored. In this article, we will answer some frequently asked questions about factoring algebraic expressions.

Q: What is factoring in algebra?

A: Factoring is a process of expressing an algebraic expression as a product of simpler expressions. It involves breaking down a complex expression into simpler components that can be multiplied together to obtain the original expression.

Q: Why is factoring important in algebra?

A: Factoring is an essential tool in algebra, as it allows us to simplify complex expressions, solve equations, and analyze functions. By factoring an expression, we can identify its roots, determine its behavior, and make predictions about its behavior.

Q: What are some common techniques for factoring algebraic expressions?

A: There are several techniques for factoring algebraic expressions, including:

• Greatest Common Factor (GCF): This involves finding the largest factor that divides each term in the expression.
• Difference of Squares: This involves factoring expressions of the form $a^2 - b^2$.
• Sum and Difference: This involves factoring expressions of the form $a^2 + b^2$ and $a^2 - b^2$.
• Grouping: This involves factoring expressions by grouping terms together.

Q: How do I determine if an expression is completely factored?

A: To determine if an expression is completely factored, you need to check if it can be further simplified by factoring. If the expression cannot be further simplified, then it is completely factored.

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

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

• Not factoring out a common factor: Make sure to factor out any common factors that exist in the expression.
• Not using the correct technique: Choose the correct technique for factoring the expression, such as GCF, Difference of Squares, or Sum and Difference.
• Not checking for errors: Double-check your work to ensure that the expression is correctly factored.

Q: How do I factor expressions with variables?

A: Factoring expressions with variables involves using the same techniques as factoring expressions with constants, but with the added complexity of variables. Make sure to factor out any common factors that exist in the expression, and use the correct technique for factoring the expression.

Q: Can I factor expressions with negative numbers?

A: Yes, you can factor expressions with negative numbers. When factoring expressions with negative numbers, make sure to consider the sign of the numbers and use the correct technique for factoring the expression.

Q: How do I factor expressions with fractions?

A: Factoring expressions with fractions involves using the same techniques as factoring expressions with whole numbers, but with the added complexity of fractions. Make sure to factor out any common factors that exist in the expression, and use the correct technique for factoring the expression.

Conclusion

In conclusion, factoring algebraic expressions is an essential tool in algebra that allows us to simplify complex expressions, solve equations, and analyze functions. By understanding the techniques for factoring algebraic expressions, we can identify the roots of an expression, determine its behavior, and make predictions about its behavior.

Key Takeaways

• Factoring is a process of expressing an algebraic expression as a product of simpler expressions.
• A completely factored expression is one that cannot be further simplified by factoring.
• There are several techniques for factoring algebraic expressions, including Greatest Common Factor (GCF), Difference of Squares, Sum and Difference, and Grouping.
• Factoring is an essential tool in algebra, as it allows us to simplify complex expressions, solve equations, and analyze functions.

Final Thoughts

Factoring algebraic expressions is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions. By understanding the techniques for factoring algebraic expressions, we can identify the roots of an expression, determine its behavior, and make predictions about its behavior.