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. 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 $4x^4 - 20x^2 - 3x^2 + 15$ by grouping.

Understanding the Polynomial

Before we can factor the polynomial, we need to understand its structure. The given polynomial is:

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

This polynomial has four terms, and we can see that there are two terms with a common factor of $x^2$. We can rewrite the polynomial as:

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

Factoring by Grouping

To factor the polynomial by grouping, we need to combine the terms in a way that allows us to factor out common factors. We can start by grouping the first two terms and the last two terms:

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

Now, we can factor out the common factor of $4x^2$ from the first group and the common factor of $-3$ from the second group:

$$4x^2\left(x^2 - 5\right) - 3\left(x^2 - 5\right)$$

Factoring the Difference of Squares

We can see that both groups have a common factor of $\left(x^2 - 5\right)$. We can factor this out to get:

$$\left(4x^2 - 3\right)\left(x^2 - 5\right)$$

Conclusion

In this article, we have shown how to factor the polynomial $4x^4 - 20x^2 - 3x^2 + 15$ by grouping. We started by combining the terms in a way that allowed us to factor out common factors, and then we factored the difference of squares to get the final result. The resulting expression is:

$$\left(4x^2 - 3\right)\left(x^2 - 5\right)$$

This is the correct answer, and it is option B.

Common Mistakes to Avoid

When factoring polynomials by grouping, there are several common mistakes to avoid. Here are a few:

• Not combining like terms: Make sure to combine like terms before factoring.
• Not factoring out common factors: Make sure to factor out common factors from each group.
• Not checking for difference of squares: Make sure to check for difference of squares when factoring.

Practice Problems

Here are a few practice problems to help you practice factoring polynomials by grouping:

• Factor the polynomial $x^4 - 9x^2 + 20$ by grouping.
• Factor the polynomial $2x^4 - 12x^2 + 10$ by grouping.
• Factor the polynomial $x^4 - 16x^2 + 64$ by grouping.

Conclusion

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Introduction

Factoring polynomials by grouping is a powerful technique that can be used to factor 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 of the most 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 combining like terms and factoring out common factors.

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

A: You should use factoring by grouping when you have a polynomial with multiple terms and you want to simplify it by factoring out common factors.

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

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

• Not combining like terms
• Not factoring out common factors
• Not checking for difference of squares

Q: How do I factor a polynomial with multiple variables?

A: To factor a polynomial with multiple variables, you can use the same technique as factoring by grouping. However, you will need to be careful to combine like terms and factor out common factors.

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

A: Yes, you can use factoring by grouping to factor polynomials with negative coefficients. However, you will need to be careful to handle the negative signs correctly.

Q: How do I check if a polynomial can be factored by grouping?

A: To check if a polynomial can be factored by grouping, you can try combining like terms and factoring out common factors. If you can simplify the polynomial in this way, then it can be factored by grouping.

Q: What are some examples of polynomials that can be factored by grouping?

A: Some examples of polynomials that can be factored by grouping include:

• $x^4 - 9x^2 + 20$
• $2x^4 - 12x^2 + 10$
• $x^4 - 16x^2 + 64$

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

A: Yes, you can use factoring by grouping to factor polynomials with fractional coefficients. However, you will need to be careful to handle the fractions correctly.

Q: How do I factor a polynomial with a binomial factor?

A: To factor a polynomial with a binomial factor, you can use the same technique as factoring by grouping. However, you will need to be careful to combine like terms and factor out common factors.

Conclusion

Factoring polynomials by grouping is a powerful technique that can be used to factor complex polynomials. By combining like terms and factoring out common factors, we can simplify the polynomial and make it easier to work with. In this article, we have answered some of the most frequently asked questions about factoring polynomials by grouping. We hope that this article has been helpful in understanding this technique.

Practice Problems

Here are a few practice problems to help you practice factoring polynomials by grouping:

• Factor the polynomial $x^4 - 9x^2 + 20$ by grouping.
• Factor the polynomial $2x^4 - 12x^2 + 10$ by grouping.
• Factor the polynomial $x^4 - 16x^2 + 64$ by grouping.

Additional Resources

If you are having trouble with factoring polynomials by grouping, there are several additional resources that you can use to help you. These include:

• Online tutorials and videos
• Practice problems and worksheets
• Online calculators and software

We hope that this article has been helpful in understanding factoring polynomials by grouping. If you have any further questions or need additional help, please don't hesitate to ask.