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 involves expressing a polynomial as a product of simpler polynomials. One of the techniques used to factor polynomials is the method of grouping, which involves grouping terms in a polynomial and factoring out common factors. In this article, we will explore how to factor the polynomial $3x^4 - 2x^2 + 15x^2 - 10$ by grouping and determine the correct factored form.

Understanding the Method of Grouping

The method of grouping involves grouping terms in a polynomial that have common factors. To factor a polynomial by grouping, we need to identify the terms that have common factors and group them together. We can then factor out the common factors from each group.

Step 1: Group the Terms

The given polynomial is $3x^4 - 2x^2 + 15x^2 - 10$. We can group the terms as follows:

• Group 1: $3x^4 - 2x^2$
• Group 2: $15x^2 - 10$

Step 2: Factor Out Common Factors

Now that we have grouped the terms, we can factor out common factors from each group.

• Group 1: $3x^4 - 2x^2$ can be factored as $x^2(3x^2 - 2)$
• Group 2: $15x^2 - 10$ can be factored as $5(3x^2 - 2)$

Step 3: Factor Out the Common Factor

Now that we have factored out common factors from each group, we can factor out the common factor from both groups.

• The common factor is $(3x^2 - 2)$
• The other factor is $x^2 + 5$

Step 4: Write the Factored Form

Now that we have factored out the common factor, we can write the factored form of the polynomial.

• The factored form is $(x^2 + 5)(3x^2 - 2)$

Conclusion

In this article, we have explored how to factor the polynomial $3x^4 - 2x^2 + 15x^2 - 10$ by grouping. We have identified the terms that have common factors, grouped them together, factored out common factors from each group, and finally factored out the common factor from both groups. The correct factored form of the polynomial is $(x^2 + 5)(3x^2 - 2)$.

Answer

The correct answer is:

• C. $(x^2 + 5)(3x^2 - 2)$

Discussion

The method of grouping is a powerful technique used to factor polynomials. By grouping terms that have common factors, we can factor out common factors from each group and finally factor out the common factor from both groups. This technique is useful in simplifying polynomials and solving equations.

Example Problems

Here are some example problems that demonstrate the method of grouping:

• Factor the polynomial $x^3 + 3x^2 + 2x + 6$ by grouping.
• Factor the polynomial $2x^4 - 5x^2 + 3x^2 - 4$ by grouping.
• Factor the polynomial $x^2 + 2x + 3x + 6$ by grouping.

Tips and Tricks

Here are some tips and tricks to help you factor polynomials by grouping:

• Identify the terms that have common factors and group them together.
• Factor out common factors from each group.
• Factor out the common factor from both groups.
• Check your answer by multiplying the factors together.

Conclusion

Q: What is the method of grouping in factoring polynomials?

A: The method of grouping involves grouping terms in a polynomial that have common factors. By grouping terms that have common factors, we can factor out common factors from each group and finally factor out the common factor from both groups.

Q: How do I identify the terms that have common factors?

A: To identify the terms that have common factors, look for terms that have the same variable or constant factor. For example, in the polynomial $3x^4 - 2x^2 + 15x^2 - 10$, the terms $-2x^2$ and $15x^2$ have a common factor of $x^2$.

Q: How do I group the terms?

A: To group the terms, identify the terms that have common factors and group them together. For example, in the polynomial $3x^4 - 2x^2 + 15x^2 - 10$, we can group the terms as follows:

• Group 1: $3x^4 - 2x^2$
• Group 2: $15x^2 - 10$

Q: How do I factor out common factors from each group?

A: To factor out common factors from each group, look for the greatest common factor (GCF) of the terms in each group. For example, in Group 1, the GCF of $3x^4$ and $-2x^2$ is $x^2$. Therefore, we can factor out $x^2$ from Group 1.

Q: How do I factor out the common factor from both groups?

A: To factor out the common factor from both groups, look for the common factor that is present in both groups. For example, in the polynomial $3x^4 - 2x^2 + 15x^2 - 10$, the common factor is $(3x^2 - 2)$. Therefore, we can factor out $(3x^2 - 2)$ from both groups.

Q: What are some common mistakes to avoid when factoring polynomials by grouping?

A: Some common mistakes to avoid when factoring polynomials by grouping include:

• Not identifying the terms that have common factors
• Not grouping the terms correctly
• Not factoring out the common factor from each group
• Not factoring out the common factor from both groups

Q: How do I check my answer when factoring polynomials by grouping?

A: To check your answer when factoring polynomials by grouping, multiply the factors together and simplify the expression. If the result is the original polynomial, then your answer is correct.

Q: What are some real-world applications of factoring polynomials by grouping?

A: Factoring polynomials by grouping has many real-world applications, including:

• Simplifying algebraic expressions
• Solving equations
• Graphing functions
• Modeling real-world phenomena

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

A: Yes, you can use factoring polynomials by grouping to solve quadratic equations. By factoring the quadratic expression, you can find the solutions to the equation.

Q: Can I use factoring polynomials by grouping to solve polynomial equations of higher degree?

A: Yes, you can use factoring polynomials by grouping to solve polynomial equations of higher degree. By factoring the polynomial expression, you can find the solutions to the equation.

Conclusion

Factoring polynomials by grouping is a powerful technique used to simplify polynomials and solve equations. By identifying the terms that have common factors, grouping them together, factoring out common factors from each group, and finally factoring out the common factor from both groups, we can factor polynomials and solve equations.