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 a fundamental concept in algebra that helps us simplify complex expressions and solve equations. One of the techniques used to factor polynomials is grouping, which involves combining terms in a way that allows us to factor out common factors. In this article, we will explore how to factor the polynomial $15x^3 - 5x^2 + 6x - 2$ by grouping.
Understanding the Polynomial
Before we start factoring, let's take a closer look at the polynomial $15x^3 - 5x^2 + 6x - 2$. This polynomial has four terms, and we can see that the first two terms have a common factor of $5x^2$, while the last two terms have a common factor of $2$. However, we need to find a way to group these terms in a way that allows us to factor out common factors.
Grouping the Terms
To factor the polynomial by grouping, we need to group the terms in a way that allows us to factor out common factors. Let's try grouping the first two terms together and the last two terms together:
Now, we can see that the first group has a common factor of $5x^2$, while the second group has a common factor of $2$. However, we still need to find a way to factor out common factors from each group.
Factoring Out Common Factors
Let's start by factoring out $5x^2$ from the first group:
Now, we can see that the first group has been factored out, and we are left with the second group:
We can see that the second group has a common factor of $2$, so we can factor it out:
Combining the Groups
Now that we have factored out common factors from each group, we can combine the groups to get the final factored form:
We can see that both groups have a common factor of $(3x - 1)$, so we can factor it out:
Conclusion
In this article, we have seen how to factor the polynomial $15x^3 - 5x^2 + 6x - 2$ by grouping. We started by grouping the terms in a way that allowed us to factor out common factors, and then we factored out common factors from each group. Finally, we combined the groups to get the final factored form. The resulting expression is $(5x^2 + 2)(3x - 1)$.
Answer
The correct answer is:
A. $(5x^2 + 2)(3x - 1)$
Discussion
This problem is a great example of how factoring by grouping can be used to simplify complex expressions. By grouping the terms in a way that allows us to factor out common factors, we can make the expression easier to work with and solve. This technique is an important tool in algebra and is used to solve a wide range of problems.
Tips and Tricks
- When factoring by grouping, it's often helpful to look for common factors within each group.
- Make sure to factor out common factors from each group before combining the groups.
- Use the distributive property to expand the expression and check your work.
Related Problems
- Factor the polynomial $12x^3 + 18x^2 - 30x - 25$ by grouping.
- Factor the polynomial $24x^3 - 36x^2 + 12x - 3$ by grouping.
- Factor the polynomial $18x^3 + 27x^2 - 9x - 3$ by grouping.
Factor $15x^3 - 5x^2 + 6x - 2$ by Grouping: Q&A ===========================================================
Q: What is factoring by grouping?
A: Factoring by grouping is a technique used to factor polynomials by grouping terms in a way that allows us to factor out common factors.
Q: How do I know which terms to group together?
A: To group terms together, look for common factors within each group. You can also try grouping the terms in a way that makes it easy to factor out common factors.
Q: What if I have a polynomial with multiple groups?
A: If you have a polynomial with multiple groups, you can factor out common factors from each group separately and then combine the groups to get the final factored form.
Q: Can I use factoring by grouping to factor any polynomial?
A: No, factoring by grouping is not suitable for all polynomials. It's best used for polynomials that can be grouped in a way that allows us to factor out common factors.
Q: How do I know if I have factored the polynomial correctly?
A: To check if you have factored the polynomial correctly, use the distributive property to expand the expression and check if it matches the original polynomial.
Q: What are some common mistakes to avoid when factoring by grouping?
A: Some common mistakes to avoid when factoring by grouping include:
- Factoring out common factors from the wrong group
- Not factoring out common factors from each group
- Not combining the groups correctly
Q: Can I use factoring by grouping to solve equations?
A: Yes, factoring by grouping can be used to solve equations. By factoring out common factors, we can make the equation easier to solve.
Q: What are some real-world applications of factoring by grouping?
A: Factoring by grouping has many real-world applications, including:
- Simplifying complex expressions in physics and engineering
- Solving equations in economics and finance
- Factoring polynomials in computer science and coding
Q: How can I practice factoring by grouping?
A: To practice factoring by grouping, try factoring polynomials with different groups and combinations of terms. You can also use online resources and practice problems to help you improve your skills.
Q: What are some tips for mastering factoring by grouping?
A: Some tips for mastering factoring by grouping include:
- Practicing regularly to build your skills and confidence
- Using online resources and practice problems to help you improve
- Breaking down complex polynomials into smaller groups and factoring out common factors
- Checking your work by expanding the expression and verifying that it matches the original polynomial
Conclusion
In this article, we have seen how to factor the polynomial $15x^3 - 5x^2 + 6x - 2$ by grouping. We have also answered some common questions about factoring by grouping and provided tips and tricks for mastering this technique. By practicing factoring by grouping, you can improve your skills and become more confident in your ability to solve complex equations and simplify complex expressions.