Select The Correct Answer.What Is The Factored Form Of 5 X − 625 X 4 5x - 625x^4 5 X − 625 X 4 ?A. 5 X ( 1 + 5 X ) ( 1 − 5 X + 25 X 2 5x(1+5x)(1-5x+25x^2 5 X ( 1 + 5 X ) ( 1 − 5 X + 25 X 2 ]B. 5 X ( 1 − 5 X ) ( 1 + 5 X + 25 X 2 5x(1-5x)(1+5x+25x^2 5 X ( 1 − 5 X ) ( 1 + 5 X + 25 X 2 ]C. ( 1 − 5 X ) ( 1 + 5 X + 25 X 2 (1-5x)(1+5x+25x^2 ( 1 − 5 X ) ( 1 + 5 X + 25 X 2 ]D. ( 1 + 5 X ) ( 1 − 5 X + 25 X 2 (1+5x)(1-5x+25x^2 ( 1 + 5 X ) ( 1 − 5 X + 25 X 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 given polynomial $5x - 625x^4$ and determine the correct factored form.

Understanding the Polynomial

Before we dive into factoring, let's take a closer look at the given polynomial $5x - 625x^4$. This polynomial consists of two terms: $5x$ and $-625x^4$. The first term is a linear term, while the second term is a quartic term.

Factoring the Polynomial

To factor the polynomial, we can start by looking for common factors. In this case, we can factor out a common factor of $5x$ from both terms:

$$5x - 625x^4 = 5x(1 - 625x^3)$$

Now, we can focus on factoring the quadratic expression $1 - 625x^3$. To do this, we can look for two numbers whose product is $-625$ and whose sum is $-1$. These numbers are $25$ and $-25$, so we can write:

$$1 - 625x^3 = (1 + 25x)(1 - 25x)$$

Now, we can substitute this expression back into the factored form of the polynomial:

$$5x - 625x^4 = 5x(1 + 25x)(1 - 25x)$$

Evaluating the Answer Choices

Now that we have factored the polynomial, let's evaluate the answer choices:

• A. $5x(1+5x)(1-5x+25x^2)$: This is not the correct factored form of the polynomial.
• B. $5x(1-5x)(1+5x+25x^2)$: This is not the correct factored form of the polynomial.
• C. $(1-5x)(1+5x+25x^2)$: This is not the correct factored form of the polynomial.
• D. $(1+5x)(1-5x+25x^2)$: This is not the correct factored form of the polynomial.

Conclusion

Based on our analysis, the correct factored form of the polynomial $5x - 625x^4$ is:

$$5x(1 + 25x)(1 - 25x)$$

This is the only answer choice that matches the factored form of the polynomial.

Tips and Tricks

When factoring polynomials, it's essential to look for common factors and to use the correct factoring techniques. Here are some tips and tricks to help you master factoring polynomials:

• Look for common factors: Before factoring, look for common factors that can be factored out from the polynomial.
• Use the correct factoring techniques: Use the correct factoring techniques, such as factoring quadratics or factoring polynomials with multiple variables.
• Check your work: Always check your work to ensure that the factored form of the polynomial is correct.

Common Mistakes to Avoid

When factoring polynomials, there are several common mistakes to avoid. Here are some of the most common mistakes:

• Factoring out the wrong factor: Make sure to factor out the correct factor from the polynomial.
• Using the wrong factoring technique: Use the correct factoring technique for the type of polynomial you are working with.
• Not checking your work: Always check your work to ensure that the factored form of the polynomial is correct.

Conclusion

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Introduction

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In our previous article, we focused on factoring the polynomial $5x - 625x^4$ and determined the correct factored form. In this article, we will provide a Q&A guide to help you master factoring polynomials.

Q&A

Q: What is factoring a polynomial?

A: Factoring a polynomial involves expressing it as a product of simpler polynomials.

Q: What are the different types of factoring?

A: There are several types of factoring, including:

• Factoring out the greatest common factor (GCF)
• Factoring quadratics
• Factoring polynomials with multiple variables
• Factoring polynomials with complex numbers

Q: How do I factor out the GCF?

A: To factor out the GCF, look for the largest factor that divides each term of the polynomial. Then, divide each term by the GCF to obtain the factored form.

Q: How do I factor quadratics?

A: To factor quadratics, look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term. Then, write the quadratic as a product of two binomials.

Q: How do I factor polynomials with multiple variables?

A: To factor polynomials with multiple variables, look for common factors and use the correct factoring techniques.

Q: How do I factor polynomials with complex numbers?

A: To factor polynomials with complex numbers, use the correct factoring techniques and consider the complex conjugate of the polynomial.

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

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

• Factoring out the wrong factor
• Using the wrong factoring technique
• Not checking your work
• Not considering the complex conjugate of the polynomial

Q: How do I check my work when factoring polynomials?

A: To check your work, multiply the factored form of the polynomial to ensure that it equals the original polynomial.

Q: What are some tips and tricks for factoring polynomials?

A: Some tips and tricks for factoring polynomials include:

• Looking for common factors
• Using the correct factoring techniques
• Checking your work
• Considering the complex conjugate of the polynomial

Conclusion

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we provided a Q&A guide to help you master factoring polynomials. By following the tips and tricks outlined in this article, you can become a proficient algebraist and master factoring polynomials.

Common Factoring Techniques

Here are some common factoring techniques:

• Factoring out the GCF: To factor out the GCF, look for the largest factor that divides each term of the polynomial. Then, divide each term by the GCF to obtain the factored form.
• Factoring quadratics: To factor quadratics, look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term. Then, write the quadratic as a product of two binomials.
• Factoring polynomials with multiple variables: To factor polynomials with multiple variables, look for common factors and use the correct factoring techniques.
• Factoring polynomials with complex numbers: To factor polynomials with complex numbers, use the correct factoring techniques and consider the complex conjugate of the polynomial.

Practice Problems

Here are some practice problems to help you master factoring polynomials:

• Factor the polynomial $x^2 + 5x + 6$
• Factor the polynomial $x^2 - 7x + 12$
• Factor the polynomial $x^3 + 2x^2 - 5x - 6$
• Factor the polynomial $x^4 - 4x^2 + 4$

Conclusion

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we provided a Q&A guide to help you master factoring polynomials. By following the tips and tricks outlined in this article, you can become a proficient algebraist and master factoring polynomials.