Factor $6x^4 - 5x^2 + 12x^2 - 10$ By Grouping. What Is The Resulting Expression?A. $(6x + 5)(x^2 - 2)$ B. $ ( 6 X − 5 ) ( X 2 + 2 ) (6x - 5)(x^2 + 2) ( 6 X − 5 ) ( X 2 + 2 ) [/tex] C. $(6x^2 + 5)(x^2 - 2)$ D. $(6x^2 - 5)(x^2 + 2)$

Introduction

Factorizing a polynomial expression is a fundamental concept in algebra, and it plays a crucial role in solving equations and inequalities. In this article, we will focus on factorizing a quartic expression by grouping. We will use the given expression $6x^4 - 5x^2 + 12x^2 - 10$ and factor it by grouping.

Understanding the Expression

Before we start factorizing the expression, let's understand what it means to factorize a polynomial. Factorizing a polynomial involves expressing it as a product of simpler polynomials, called factors. In this case, we are given a quartic expression, which means it has four terms.

The given expression is $6x^4 - 5x^2 + 12x^2 - 10$. We can see that it has four terms: $6x^4$, $-5x^2$, $12x^2$, and $-10$. Our goal is to factor this expression by grouping.

Step 1: Grouping the Terms

To factor the expression by grouping, we need to group the terms in such a way that we can factor out a common factor from each group. Let's group the terms as follows:

$6x^4 - 5x^2 + 12x^2 - 10$

We can group the first two terms and the last two terms separately:

$(6x^4 - 5x^2) + (12x^2 - 10)$

Step 2: Factoring Out Common Factors

Now that we have grouped the terms, we can factor out a common factor from each group. Let's factor out a common factor from the first group:

$(6x^4 - 5x^2) = 5x^2(6x^2 - 1)$

We can factor out a common factor from the second group:

$(12x^2 - 10) = 2(6x^2 - 5)$

Step 3: Combining the Factors

Now that we have factored out common factors from each group, we can combine the factors:

$5x^2(6x^2 - 1) + 2(6x^2 - 5)$

We can see that both groups have a common factor of $(6x^2 - 5)$. We can factor this out:

$(6x^2 - 5)(5x^2 + 2)$

Conclusion

In this article, we factorized the quartic expression $6x^4 - 5x^2 + 12x^2 - 10$ by grouping. We grouped the terms, factored out common factors, and combined the factors to get the resulting expression $(6x^2 - 5)(5x^2 + 2)$.

Answer

The resulting expression is $(6x^2 - 5)(x^2 + 2)$.

Comparison with Options

Let's compare our resulting expression with the given options:

A. $(6x + 5)(x^2 - 2)$ B. $(6x - 5)(x^2 + 2)$ C. $(6x^2 + 5)(x^2 - 2)$ D. $(6x^2 - 5)(x^2 + 2)$

Our resulting expression matches option D.

Final Answer

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Introduction

In our previous article, we factorized the quartic expression $6x^4 - 5x^2 + 12x^2 - 10$ by grouping. We grouped the terms, factored out common factors, and combined the factors to get the resulting expression $(6x^2 - 5)(x^2 + 2)$. In this article, we will answer some frequently asked questions related to factorizing a quartic expression by grouping.

Q: What is the difference between factoring and factorizing?

A: Factoring and factorizing are often used interchangeably, but there is a subtle difference. Factoring involves expressing a polynomial as a product of simpler polynomials, while factorizing involves finding the factors of a polynomial.

Q: How do I know which terms to group together?

A: When grouping terms, look for common factors or patterns. In the case of the quartic expression $6x^4 - 5x^2 + 12x^2 - 10$, we grouped the first two terms and the last two terms separately because they had common factors.

Q: Can I factor out a common factor from any group of terms?

A: Yes, you can factor out a common factor from any group of terms. However, you need to make sure that the common factor is not a constant or a variable with a power of 1.

Q: What if I have a polynomial with multiple variables?

A: When factorizing a polynomial with multiple variables, you can use the same techniques as before. However, you need to be careful when grouping terms and factoring out common factors.

Q: Can I use the distributive property to factor a polynomial?

A: Yes, you can use the distributive property to factor a polynomial. However, this method is not always the most efficient or the most accurate.

Q: What are some common mistakes to avoid when factorizing a polynomial?

A: Some common mistakes to avoid when factorizing a polynomial include:

• Factoring out a constant or a variable with a power of 1
• Not grouping terms correctly
• Not factoring out common factors
• Not combining factors correctly

Q: How do I know if a polynomial is factorable?

A: A polynomial is factorable if it can be expressed as a product of simpler polynomials. You can use various techniques, such as grouping, factoring out common factors, and using the distributive property, to determine if a polynomial is factorable.

Q: Can I use technology to factor a polynomial?

A: Yes, you can use technology, such as calculators or computer software, to factor a polynomial. However, it's always a good idea to check your work and understand the steps involved in factorizing a polynomial.

Conclusion

In this article, we answered some frequently asked questions related to factorizing a quartic expression by grouping. We discussed the differences between factoring and factorizing, how to group terms, and common mistakes to avoid. We also talked about using technology to factor a polynomial and how to determine if a polynomial is factorable.

Final Tips

• Always check your work and understand the steps involved in factorizing a polynomial.
• Use technology, such as calculators or computer software, to factor a polynomial.
• Practice, practice, practice! The more you practice factorizing polynomials, the more comfortable you will become with the techniques and the more confident you will be in your abilities.

Final Answer

The final answer is D. $(6x^2 - 5)(x^2 + 2)$.