Factor $15x^3 - 5x^2 + 6x - 2$ By Grouping. What Is The Resulting Expression?A. $(5x^2 + 2)(3x - 1)$ B. $ ( 5 X 2 − 2 ) ( 3 X + 1 ) (5x^2 - 2)(3x + 1) ( 5 X 2 − 2 ) ( 3 X + 1 ) [/tex] C. $(15x^2 + 2)(x - 1)$ D. $(15x^2 - 2)(x + 1)$
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 explore how to factor the polynomial $15x^3 - 5x^2 + 6x - 2$ by grouping, and we will also examine the different options provided in the multiple-choice questions.
What is Factoring by Grouping?
Factoring by grouping is a method used to factor polynomials that cannot be factored using the greatest common factor (GCF) or the difference of squares. This method involves grouping the terms of the polynomial in a way that allows us to factor out common factors from each group.
Step 1: Group the Terms
To factor the polynomial $15x^3 - 5x^2 + 6x - 2$ by grouping, we need to group the terms in a way that allows us to factor out common factors from each group. We can group the terms as follows:
Step 2: Factor Out Common Factors
Now that we have grouped the terms, we can factor out common factors from each group. We can factor out $3x$ from the first group and $-1$ from the second group:
Step 3: Factor Out the Common Binomial
Now that we have factored out common factors from each group, we can factor out the common binomial $(5x^2 + 2)$ from both groups:
Conclusion
In conclusion, we have factored the polynomial $15x^3 - 5x^2 + 6x - 2$ by grouping, and we have obtained the resulting expression $(3x - 1)(5x^2 + 2)$. This expression is one of the options provided in the multiple-choice questions.
Comparison with Other Options
Let's compare our resulting expression with the other options provided in the multiple-choice questions:
- Option A: $(5x^2 + 2)(3x - 1)$
- Option B: $(5x^2 - 2)(3x + 1)$
- Option C: $(15x^2 + 2)(x - 1)$
- Option D: $(15x^2 - 2)(x + 1)$
Our resulting expression $(3x - 1)(5x^2 + 2)$ is different from all the other options. Therefore, the correct answer is:
- A. $(5x^2 + 2)(3x - 1)$
However, we must note that the correct answer is actually $(3x - 1)(5x^2 + 2)$, not $(5x^2 + 2)(3x - 1)$. This is because the order of the factors does not matter when we are factoring by grouping.
Final Answer
Q: What is factoring by grouping?
A: Factoring by grouping is a method used to factor polynomials that cannot be factored using the greatest common factor (GCF) or the difference of squares. This method involves grouping the terms of the polynomial in a way that allows us to factor out common factors from each group.
Q: How do I know which terms to group together?
A: To group the terms, look for common factors or patterns within the polynomial. You can also try grouping the terms in different ways to see if any of them work.
Q: What if I have a polynomial with multiple variables?
A: Factoring by grouping can still be used with polynomials that have multiple variables. However, you may need to use additional techniques, such as the distributive property, to simplify the polynomial before factoring by grouping.
Q: Can I use factoring by grouping with polynomials that have negative coefficients?
A: Yes, factoring by grouping can be used with polynomials that have negative coefficients. However, you may need to use additional techniques, such as multiplying both sides of the equation by -1, to simplify the polynomial before factoring by grouping.
Q: How do I know if I have factored the polynomial correctly?
A: To check if you have factored the polynomial correctly, multiply the factors together and simplify the expression. If the result is the original polynomial, then you have factored it correctly.
Q: What if I get a different answer than the one provided in the multiple-choice questions?
A: If you get a different answer than the one provided in the multiple-choice questions, it may be because the order of the factors is different. Remember that the order of the factors does not matter when we are factoring by grouping.
Q: Can I use factoring by grouping with polynomials that have rational coefficients?
A: Yes, factoring by grouping can be used with polynomials that have rational coefficients. However, you may need to use additional techniques, such as the rational root theorem, to simplify the polynomial before factoring by grouping.
Q: How do I know if a polynomial can be factored by grouping?
A: A polynomial can be factored by grouping if it can be written in the form:
or
If the polynomial can be written in one of these forms, then it can be factored 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 grouping the terms correctly
- Not factoring out the common factors correctly
- Not checking the result to make sure it is the original polynomial
- Not using the correct order of the factors
Conclusion
Factoring by grouping is a powerful technique for factoring polynomials that cannot be factored using the greatest common factor (GCF) or the difference of squares. By following the steps outlined in this article and avoiding common mistakes, you can master the art of factoring by grouping and become a proficient mathematician.
Additional Resources
For more information on factoring by grouping, check out the following resources:
- Khan Academy: Factoring by Grouping
- Mathway: Factoring by Grouping
- Wolfram Alpha: Factoring by Grouping
Remember, practice makes perfect! Try factoring by grouping with different polynomials to get a feel for the technique.