Factor The Polynomial $4x^4 - 20x^2 - 3x^2 + 15$ By Grouping. What Is The Resulting Expression?A. $\left(4x^2 + 3\right)\left(x^2 - 5\right)$B. $ ( 4 X 2 − 3 ) ( X 2 − 5 ) \left(4x^2 - 3\right)\left(x^2 - 5\right) ( 4 X 2 − 3 ) ( X 2 − 5 ) [/tex]C. $\left(4x^2 -

Introduction

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we will focus on factoring the polynomial $4x^4 - 20x^2 - 3x^2 + 15$ by grouping. This technique is useful for simplifying complex polynomials and can be applied to a wide range of mathematical problems.

Understanding the Polynomial

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

$$4x^4 - 20x^2 - 3x^2 + 15$$

This polynomial consists of four terms, each with a different degree. To factor this polynomial by grouping, we need to identify the common factors among the terms.

Step 1: Identify Common Factors

The first step in factoring by grouping is to identify the common factors among the terms. In this case, we can see that the terms $-20x^2$ and $-3x^2$ have a common factor of $-x^2$. We can rewrite the polynomial as:

$$4x^4 - x^2(20 + 3) + 15$$

This simplifies to:

$$4x^4 - x^2(23) + 15$$

Step 2: Group the Terms

Now that we have identified the common factor, we can group the terms accordingly. We can group the first two terms, which have a common factor of $4x^4$, and the last two terms, which have a common factor of $-x^2(23)$:

$$4x^4 - x^2(23) + 15$$

Step 3: Factor the Groups

Now that we have grouped the terms, we can factor each group separately. The first group, $4x^4$, can be factored as:

$$4x^4 = (2x^2)^2$$

The second group, $-x^2(23)$, can be factored as:

$$-x^2(23) = -(23x^2)$$

The third group, $15$, cannot be factored further.

Step 4: Combine the Factors

Now that we have factored each group, we can combine the factors to obtain the final result:

$$(2x^2)^2 - (23x^2) + 15$$

This can be rewritten as:

$$(2x^2 - 1)(2x^2 - 15)$$

However, this is not the correct answer. We need to factor the polynomial in a different way.

Alternative Factoring Method

Let's try an alternative factoring method. We can start by factoring out the greatest common factor (GCF) of the polynomial, which is $1$. This gives us:

$$4x^4 - 20x^2 - 3x^2 + 15$$

We can then group the terms as follows:

$$(4x^4 - 20x^2) - (3x^2 + 15)$$

This can be rewritten as:

$$4x^2(x^2 - 5) - 3(x^2 + 5)$$

We can then factor out the common factor of $4x^2 - 3$:

$$(4x^2 - 3)(x^2 - 5)$$

This is the correct answer.

Conclusion

In this article, we have shown how to factor the polynomial $4x^4 - 20x^2 - 3x^2 + 15$ by grouping. We have also presented an alternative factoring method that yields the correct answer. Factoring polynomials by grouping is a useful technique that can be applied to a wide range of mathematical problems.

Final Answer

The final answer is:

(4x^2 - 3)(x^2 - 5)$<br/> **Factoring Polynomials by Grouping: A Q&A Guide** =====================================================

Introduction

Factoring polynomials by grouping is a powerful technique used to simplify complex polynomials. In our previous article, we showed how to factor the polynomial $4x^4 - 20x^2 - 3x^2 + 15$ by grouping. 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. This technique is useful for simplifying complex polynomials and can be applied to a wide range of mathematical problems.

Q: How do I identify common factors in a polynomial?

A: To identify common factors in a polynomial, look for terms that have a common factor. This can be a numerical factor, a variable factor, or a combination of both. For example, in the polynomial $4x^4 - 20x^2 - 3x^2 + 15$, the terms $-20x^2$ and $-3x^2$ have a common factor of $-x^2$.

Q: How do I group terms in a polynomial?

A: To group terms in a polynomial, look for terms that have a common factor. Group these terms together and factor out the common factor. For example, in the polynomial $4x^4 - 20x^2 - 3x^2 + 15$, we can group the terms $-20x^2$ and $-3x^2$ together and factor out the common factor of $-x^2$.

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

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

• Not identifying common factors in the polynomial
• Not grouping terms correctly
• Not factoring out the common factor correctly
• Not checking for other factoring methods

Q: Can I use factoring by grouping to factor all polynomials?

A: No, factoring by grouping is not a universal factoring method. It is only useful for polynomials that can be grouped into terms with common factors. Other factoring methods, such as factoring by difference of squares or factoring by grouping, may be more suitable for other types of polynomials.

Q: How do I know when to use factoring by grouping?

A: You should use factoring by grouping when:

• The polynomial has terms with common factors
• The polynomial is complex and difficult to factor using other methods
• You want to simplify the polynomial and make it easier to work with

Q: Can I use factoring by grouping to factor polynomials with negative coefficients?

A: Yes, factoring by grouping can be used to factor polynomials with negative coefficients. However, you may need to use other factoring methods, such as factoring by difference of squares, to factor the polynomial.

Q: Can I use factoring by grouping to factor polynomials with fractional coefficients?

A: Yes, factoring by grouping can be used to factor polynomials with fractional coefficients. However, you may need to use other factoring methods, such as factoring by difference of squares, to factor the polynomial.

Conclusion

In this article, we have answered some frequently asked questions about factoring polynomials by grouping. We have also provided some tips and tricks for using this technique effectively. Factoring by grouping is a powerful technique that can be used to simplify complex polynomials and make them easier to work with.

Final Tips

• Always identify common factors in the polynomial before grouping terms
• Group terms carefully to ensure that you are factoring out the correct common factor
• Check for other factoring methods, such as factoring by difference of squares, before using factoring by grouping
• Practice, practice, practice! The more you practice factoring by grouping, the more comfortable you will become with this technique.