Complete The Steps To Factor 12 X 3 − 9 X 2 + 8 X − 6 12x^3 - 9x^2 + 8x - 6 12 X 3 − 9 X 2 + 8 X − 6 By Grouping.Step 1: Group The First Two Terms And The Second Two Terms.Step 2: What Is The Greatest Common Factor Of Group 1?A. 3 X 3x 3 X B. 6 X 6x 6 X C. 3 X 2 3x^2 3 X 2

Introduction

Factoring polynomials is an essential skill in algebra, and one of the most effective methods for factoring is by grouping. In this article, we will guide you through the steps to factor the polynomial $12x^3 - 9x^2 + 8x - 6$ by grouping. We will break down the process into manageable steps, making it easy to understand and apply.

Step 1: Group the First Two Terms and the Second Two Terms

To factor the polynomial by grouping, we need to group the first two terms and the second two terms. This means we will separate the polynomial into two groups: Group 1 and Group 2.

Group 1: $12x^3 - 9x^2$ Group 2: $8x - 6$

Step 2: Factor Out the Greatest Common Factor (GCF) of Group 1

Now that we have grouped the terms, we need to find the greatest common factor (GCF) of Group 1. The GCF is the largest expression that divides each term in the group without leaving a remainder.

# Greatest Common Factor (GCF) of Group 1
## Factoring Out the GCF
To find the GCF of Group 1, we need to identify the common factors of the two terms.

The first term is $12x^3$, which has factors of $3$, $4$, $x$, and $x^2$.
The second term is $-9x^2$, which has factors of $-3$, $3$, $x$, and $x^2$.

The common factors of the two terms are $-3$, $x$, and $x^2$. Therefore, the GCF of Group 1 is $-3x^2$.
Factoring Out the GCF
Now that we have found the GCF, we can factor it out of Group 1.
$12x^3 - 9x^2 = -3x^2(4x - 3)$
Conclusion
We have successfully factored out the GCF of Group 1, which is $-3x^2$.

Step 3: Factor Out the Greatest Common Factor (GCF) of Group 2

Now that we have factored out the GCF of Group 1, we need to find the GCF of Group 2. The GCF is the largest expression that divides each term in the group without leaving a remainder.

# Greatest Common Factor (GCF) of Group 2
## Factoring Out the GCF
To find the GCF of Group 2, we need to identify the common factors of the two terms.

The first term is $8x$, which has factors of $2$, $2$, $x$, and $x$.
The second term is $-6$, which has factors of $-2$, $3$, and $x$.

The common factors of the two terms are $-2$ and $x$. Therefore, the GCF of Group 2 is $-2x$.
Factoring Out the GCF
Now that we have found the GCF, we can factor it out of Group 2.
$8x - 6 = -2x(4 - 3)$
Conclusion
We have successfully factored out the GCF of Group 2, which is $-2x$.

Step 4: Combine the Factored Groups

Now that we have factored out the GCF of both groups, we can combine the factored groups to get the final factored form of the polynomial.

# Combining the Factored Groups
## Final Factored Form
We have factored out the GCF of Group 1, which is $-3x^2$, and the GCF of Group 2, which is $-2x$.
$12x^3 - 9x^2 + 8x - 6 = -3x^2(4x - 3) - 2x(4 - 3)$
Simplifying the Expression
We can simplify the expression by combining like terms.
$-3x^2(4x - 3) - 2x(4 - 3) = -3x^2(4x - 3) - 2x(-1)$
Final Factored Form
The final factored form of the polynomial is:
$12x^3 - 9x^2 + 8x - 6 = -3x^2(4x - 3) + 2x$
Conclusion
We have successfully factored the polynomial $12x^3 - 9x^2 + 8x - 6$ by grouping.

Conclusion

Factoring polynomials by grouping is a powerful technique that can be used to factor complex polynomials. By following the steps outlined in this article, you can factor polynomials with ease. Remember to group the terms, find the GCF of each group, and combine the factored groups to get the final factored form of the polynomial.

Discussion

What is the greatest common factor of Group 1?

A. $3x$ B. $6x$ C. $3x^2$

Introduction

Factoring polynomials by grouping is a powerful technique that can be used to factor complex polynomials. In our previous article, we provided a step-by-step guide on how to factor the polynomial $12x^3 - 9x^2 + 8x - 6$ by grouping. In this article, we will answer some of the most frequently asked questions about factoring polynomials by grouping.

Q: What is the greatest common factor (GCF) of Group 1?

A: The greatest common factor (GCF) of Group 1 is $-3x^2$. This is the largest expression that divides each term in Group 1 without leaving a remainder.

Q: How do I find the GCF of a group of terms?

A: To find the GCF of a group of terms, you need to identify the common factors of each term. The common factors are the factors that are shared by all the terms in the group. Once you have identified the common factors, you can multiply them together to get the GCF.

Q: What is the difference between factoring by grouping and factoring by greatest common factor (GCF)?

A: Factoring by grouping and factoring by greatest common factor (GCF) are two different techniques used to factor polynomials. Factoring by grouping involves grouping the terms of a polynomial into two or more groups and then factoring out the greatest common factor (GCF) of each group. Factoring by GCF involves finding the greatest common factor (GCF) of all the terms in the polynomial and then factoring it out.

Q: Can I factor a polynomial by grouping if it has more than two terms?

A: Yes, you can factor a polynomial by grouping if it has more than two terms. However, you need to group the terms in a way that makes it easy to find the greatest common factor (GCF) of each group.

Q: What is the final factored form of the polynomial $12x^3 - 9x^2 + 8x - 6$?

A: The final factored form of the polynomial $12x^3 - 9x^2 + 8x - 6$ is:

$12x^3 - 9x^2 + 8x - 6 = -3x^2(4x - 3) + 2x$

Q: How do I know if a polynomial can be factored by grouping?

A: A polynomial can be factored by grouping if it has a greatest common factor (GCF) that can be factored out of each group of terms.

Q: What are some common mistakes to avoid when factoring polynomials by grouping?

A: Some common mistakes to avoid when factoring polynomials by grouping include:

• Not grouping the terms correctly
• Not finding the greatest common factor (GCF) of each group
• Not factoring out the greatest common factor (GCF) of each group
• Not combining the factored groups correctly

Conclusion

Factoring polynomials by grouping is a powerful technique that can be used to factor complex polynomials. By following the steps outlined in this article, you can factor polynomials with ease. Remember to group the terms, find the GCF of each group, and combine the factored groups to get the final factored form of the polynomial.

Discussion

Do you have any questions about factoring polynomials by grouping? Ask us in the comments below!

Additional Resources

• [Factoring Polynomials by Grouping: A Step-by-Step Guide](link to previous article)
• [Factoring Polynomials by Greatest Common Factor (GCF)](link to resource)
• [Polynomial Factoring Techniques](link to resource)