Complete The Steps To Factor $12x^3 - 9x^2 + 8x - 6$ By Grouping.Step 1: Group The First Two Terms And The Second Two Terms.
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Introduction
Factoring polynomials is an essential skill in algebra, and it can be a challenging task, especially when dealing with cubic polynomials. In this article, we will focus on factoring the cubic polynomial $12x^3 - 9x^2 + 8x - 6$ by grouping. This method involves grouping the terms of the polynomial in a way that allows us to factor out common factors.
Step 1: Group the First Two Terms and the Second Two Terms
To factor the given polynomial by grouping, we need to group the first two terms and the second two terms. The polynomial is $12x^3 - 9x^2 + 8x - 6.$ We can group the first two terms as follows:
And the second two terms can be grouped as follows:
Step 2: Factor Out Common Factors
Now that we have grouped the terms, we can factor out common factors. We can see that both groups have a common factor of . We can factor this out as follows:
Step 3: Factor Out the Common Binomial Factor
Now that we have factored out the common factor of , we can factor out the common binomial factor. We can see that both groups have a common binomial factor of . We can factor this out as follows:
Conclusion
In this article, we have shown how to factor the cubic polynomial $12x^3 - 9x^2 + 8x - 6$ by grouping. We grouped the first two terms and the second two terms, factored out common factors, and finally factored out the common binomial factor. This method is useful for factoring polynomials that have a common binomial factor.
Tips and Tricks
- When factoring by grouping, make sure to group the terms in a way that allows you to factor out common factors.
- Use the distributive property to expand the groups and factor out common factors.
- Check your work by multiplying the factors together to make sure you get the original polynomial.
Common Mistakes to Avoid
- Don't forget to factor out common factors when grouping the terms.
- Make sure to use the distributive property to expand the groups and factor out common factors.
- Check your work by multiplying the factors together to make sure you get the original polynomial.
Real-World Applications
Factoring polynomials by grouping has many real-world applications. For example, it can be used to solve systems of equations, find the roots of a polynomial, and optimize functions. It is also used in many fields such as physics, engineering, and economics.
Conclusion
In conclusion, factoring a cubic polynomial by grouping is a useful technique that can be used to factor polynomials that have a common binomial factor. By following the steps outlined in this article, you can factor the polynomial $12x^3 - 9x^2 + 8x - 6$ and many others like it. Remember to group the terms in a way that allows you to factor out common factors, use the distributive property to expand the groups, and check your work by multiplying the factors together.
Final Answer
The final answer is:
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Introduction
In our previous article, we showed how to factor the cubic polynomial $12x^3 - 9x^2 + 8x - 6$ by grouping. This method involves grouping the terms of the polynomial in a way that allows us to factor out common factors. In this article, we will answer some common questions that students may have when factoring polynomials by grouping.
Q: What is factoring by grouping?
A: Factoring by grouping is a method of factoring polynomials that involves grouping the terms of the polynomial in a way that allows us to factor out common factors.
Q: How do I know which terms to group together?
A: When grouping the terms, you should group the terms that have the most common factors. In the case of the polynomial $12x^3 - 9x^2 + 8x - 6$, we grouped the first two terms and the second two terms because they both had a common factor of .
Q: What if I don't see a common factor?
A: If you don't see a common factor, you may need to try different groupings or use a different factoring method. It's also possible that the polynomial cannot be factored further.
Q: Can I use factoring by grouping to factor any polynomial?
A: No, factoring by grouping is not suitable for all polynomials. This method is best used for polynomials that have a common binomial factor.
Q: How do I know if a polynomial can be factored by grouping?
A: To determine if a polynomial can be factored by grouping, look for a common binomial factor. If you can find a common binomial factor, you can try factoring the polynomial by grouping.
Q: What are some common mistakes to avoid when factoring by grouping?
A: Some common mistakes to avoid when factoring by grouping include:
- Not factoring out common factors
- Not using the distributive property to expand the groups
- Not checking your work by multiplying the factors together
Q: How do I check my work when factoring by grouping?
A: To check your work when factoring by grouping, multiply the factors together to make sure you get the original polynomial.
Q: Can I use factoring by grouping to solve systems of equations?
A: Yes, factoring by grouping can be used to solve systems of equations. By factoring the polynomial, you can find the roots of the polynomial, which can be used to solve the system of equations.
Q: Can I use factoring by grouping to optimize functions?
A: Yes, factoring by grouping can be used to optimize functions. By factoring the polynomial, you can find the maximum or minimum value of the function.
Conclusion
In conclusion, factoring a cubic polynomial by grouping is a useful technique that can be used to factor polynomials that have a common binomial factor. By following the steps outlined in this article, you can factor the polynomial $12x^3 - 9x^2 + 8x - 6$ and many others like it. Remember to group the terms in a way that allows you to factor out common factors, use the distributive property to expand the groups, and check your work by multiplying the factors together.
Final Answer
The final answer is: