Factor The Polynomial 3 X 4 − 2 X 2 + 15 X 2 − 10 3x^4 - 2x^2 + 15x^2 - 10 3 X 4 − 2 X 2 + 15 X 2 − 10 By Grouping. Which Product Is The Factored Form Of The Polynomial?A. ( − X 2 − 5 ) ( 3 X 2 + 2 (-x^2 - 5)(3x^2 + 2 ( − X 2 − 5 ) ( 3 X 2 + 2 ]B. ( X 2 − 2 ) ( 3 X 2 + 5 (x^2 - 2)(3x^2 + 5 ( X 2 − 2 ) ( 3 X 2 + 5 ]C. ( X 2 + 5 ) ( 3 X 2 − 2 (x^2 + 5)(3x^2 - 2 ( X 2 + 5 ) ( 3 X 2 − 2 ]D. $(3x^2 - 5)(x^2 +

Introduction

Factoring polynomials is a fundamental concept in algebra that helps us simplify complex expressions and solve equations. In this article, we will focus on factoring the polynomial $3x^4 - 2x^2 + 15x^2 - 10$ by grouping. This technique involves grouping terms that have common factors and then factoring out those common factors. We will explore the different options and determine which product is the factored form of the polynomial.

Understanding the Polynomial

Before we begin factoring, let's take a closer look at the given polynomial:

$3x^4 - 2x^2 + 15x^2 - 10$

We can see that the polynomial has four terms, and each term has a different power of $x$. To factor this polynomial, we need to identify the common factors among the terms.

Step 1: Grouping Terms

The first step in factoring by grouping is to group the terms that have common factors. In this case, we can group the first two terms and the last two terms:

$(3x^4 - 2x^2) + (15x^2 - 10)$

Now, let's examine each group separately.

Group 1: $3x^4 - 2x^2$

The first group has two terms: $3x^4$ and $-2x^2$. We can factor out the common factor $x^2$ from both terms:

$x^2(3x^2 - 2)$

Group 2: $15x^2 - 10$

The second group has two terms: $15x^2$ and $-10$. We can factor out the common factor $5$ from both terms:

$5(3x^2 - 2)$

Step 2: Factoring Out Common Factors

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

$x^2(3x^2 - 2) + 5(3x^2 - 2)$

We can see that both groups have a common factor of $(3x^2 - 2)$. We can factor this out to get:

$(x^2 + 5)(3x^2 - 2)$

Conclusion

In conclusion, the factored form of the polynomial $3x^4 - 2x^2 + 15x^2 - 10$ by grouping is:

$(x^2 + 5)(3x^2 - 2)$

This is the correct answer among the options provided.

Comparison with Options

Let's compare our result with the options provided:

A. $(-x^2 - 5)(3x^2 + 2)$ B. $(x^2 - 2)(3x^2 + 5)$ C. $(x^2 + 5)(3x^2 - 2)$ D. $(3x^2 - 5)(x^2 + 2)$

We can see that option C matches our result exactly.

Final Answer

The final answer is:

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Introduction

In our previous article, we explored the concept of factoring polynomials by grouping. We learned how to factor the polynomial $3x^4 - 2x^2 + 15x^2 - 10$ by grouping and identified the correct factored form. In this article, we will answer some frequently asked questions about factoring polynomials by grouping.

Q: What is factoring by grouping?

A: Factoring by grouping is a technique used to factor polynomials by grouping terms that have common factors and then factoring out those common factors.

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

A: To group terms together, look for common factors among the terms. You can group terms that have the same power of $x$ or terms that have a common numerical factor.

Q: What if I have a polynomial with multiple groups of terms?

A: If you have a polynomial with multiple groups of terms, you can factor out the common factors from each group separately and then combine the groups.

Q: Can I always factor a polynomial by grouping?

A: No, not all polynomials can be factored by grouping. Some polynomials may not have any common factors among the terms, or they may have complex factors that cannot be factored further.

Q: How do I know if I have factored a polynomial correctly?

A: To check if you have factored a polynomial correctly, multiply the factors together and see if you get the original polynomial. If you do, then you have factored it correctly.

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 a term that is not a common factor among the terms
• Not factoring out a common factor that is present among the terms
• Not checking if the factors multiply together to give the original polynomial

Q: Can I use factoring by grouping to solve equations?

A: Yes, factoring by grouping can be used to solve equations. By factoring the polynomial on one side of the equation, you can isolate the variable and solve for its value.

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
• Modeling population growth and decay in biology and ecology

Conclusion

In conclusion, factoring by grouping is a powerful technique for simplifying complex polynomials and solving equations. By understanding the basics of factoring by grouping, you can apply this technique to a wide range of problems in mathematics and other fields.

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

Final Answer

The final answer is: There is no final numerical answer to this article, as it is a Q&A article. However, the correct factored form of the polynomial $3x^4 - 2x^2 + 15x^2 - 10$ is:

$(x^2 + 5)(3x^2 - 2)$