Which Product Of Prime Polynomials Is Equivalent To $8x^4 + 36x^3 - 72x^2$?A. $4x(2x-3)(x^2+6$\] B. $4x^2(2x-3)(x+6$\] C. $2x(2x-3)(2x^2+6$\] D. $2x(2x+3)(x^2-6$\]

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Introduction

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. This technique is essential in solving equations, graphing functions, and simplifying expressions. In this article, we will focus on factoring prime polynomials and finding equivalent products.

What are Prime Polynomials?

A prime polynomial is a polynomial that cannot be factored into the product of simpler polynomials. In other words, it is a polynomial that has no common factors other than 1. For example, the polynomial x2+1x^2 + 1 is a prime polynomial because it cannot be factored into the product of simpler polynomials.

The Problem

We are given the polynomial 8x4+36x3−72x28x^4 + 36x^3 - 72x^2 and asked to find the product of prime polynomials that is equivalent to it. To solve this problem, we need to factor the given polynomial and then identify the equivalent product.

Step 1: Factor Out the Greatest Common Factor (GCF)

The first step in factoring the polynomial is to factor out the greatest common factor (GCF). In this case, the GCF is 4x^2. Factoring out the GCF, we get:

8x4+36x3−72x2=4x2(2x2+9x−18)8x^4 + 36x^3 - 72x^2 = 4x^2(2x^2 + 9x - 18)

Step 2: Factor the Quadratic Expression

The next step is to factor the quadratic expression 2x2+9x−182x^2 + 9x - 18. To do this, we need to find two numbers whose product is -36 and whose sum is 9. These numbers are 12 and -3. Therefore, we can factor the quadratic expression as:

2x2+9x−18=(2x−3)(x+6)2x^2 + 9x - 18 = (2x - 3)(x + 6)

Step 3: Identify the Equivalent Product

Now that we have factored the polynomial, we can identify the equivalent product. The factored form of the polynomial is:

8x4+36x3−72x2=4x2(2x−3)(x+6)8x^4 + 36x^3 - 72x^2 = 4x^2(2x - 3)(x + 6)

Comparing this with the given options, we can see that the equivalent product is:

4x2(2x−3)(x+6)4x^2(2x - 3)(x + 6)

Conclusion

In this article, we have demonstrated how to factor a polynomial and find the equivalent product. We have also identified the correct answer among the given options. Factoring polynomials is an essential skill in algebra, and with practice, you can become proficient in factoring even the most complex polynomials.

Final Answer

The final answer is:

  • B. 4x2(2x−3)(x+6)4x^2(2x-3)(x+6)

This answer is based on the factored form of the polynomial, which is 4x2(2x−3)(x+6)4x^2(2x - 3)(x + 6). This is the equivalent product of prime polynomials that is equivalent to the given polynomial 8x4+36x3−72x28x^4 + 36x^3 - 72x^2.

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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 demonstrated how to factor a polynomial and find the equivalent product. In this article, we will provide a Q&A guide to help you understand the concept of factoring polynomials better.

Q: What is factoring a polynomial?

A: Factoring a polynomial involves expressing it as a product of simpler polynomials. This means that you need to break down the polynomial into its constituent parts, which are the factors.

Q: Why is factoring a polynomial important?

A: Factoring a polynomial is important because it helps you to simplify complex expressions, solve equations, and graph functions. It is also essential in algebraic manipulations, such as solving systems of equations and finding the roots of a polynomial.

Q: What are the different types of factoring?

A: There are several types of factoring, including:

  • Greatest Common Factor (GCF) factoring: This involves factoring out the greatest common factor of the terms in the polynomial.
  • Difference of Squares factoring: This involves factoring the difference of two squares, which is a polynomial of the form a2−b2a^2 - b^2.
  • Sum and Difference of Cubes factoring: This involves factoring the sum or difference of two cubes, which is a polynomial of the form a3+b3a^3 + b^3 or a3−b3a^3 - b^3.
  • Quadratic factoring: This involves factoring a quadratic expression, which is a polynomial of the form ax2+bx+cax^2 + bx + c.

Q: How do I factor a polynomial?

A: To factor a polynomial, you need to follow these steps:

  1. Check for GCF: Check if there is a greatest common factor that can be factored out of the terms in the polynomial.
  2. Look for difference of squares: Check if the polynomial can be factored as the difference of two squares.
  3. Look for sum and difference of cubes: Check if the polynomial can be factored as the sum or difference of two cubes.
  4. Use quadratic formula: If the polynomial is a quadratic expression, use the quadratic formula to find the roots.
  5. Check for other factoring patterns: Check if the polynomial can be factored using other factoring patterns, such as factoring by grouping.

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

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

  • Not checking for GCF: Failing to check for the greatest common factor can lead to incorrect factoring.
  • Not looking for difference of squares: Failing to look for the difference of squares can lead to incorrect factoring.
  • Not using the quadratic formula: Failing to use the quadratic formula when factoring a quadratic expression can lead to incorrect factoring.
  • Not checking for other factoring patterns: Failing to check for other factoring patterns can lead to incorrect factoring.

Q: How can I practice factoring polynomials?

A: You can practice factoring polynomials by:

  • Solving problems: Practice solving problems that involve factoring polynomials.
  • Using online resources: Use online resources, such as factoring calculators and worksheets, to practice factoring polynomials.
  • Working with a tutor: Work with a tutor who can provide guidance and support as you practice factoring polynomials.

Conclusion

Factoring polynomials is an essential skill in algebra that involves expressing a polynomial as a product of simpler polynomials. By following the steps outlined in this article, you can improve your factoring skills and become proficient in factoring even the most complex polynomials. Remember to practice regularly and seek help when needed to master this skill.

Final Tips

  • Practice regularly: Practice factoring polynomials regularly to improve your skills.
  • Seek help when needed: Seek help from a tutor or online resources when you need assistance with factoring polynomials.
  • Use online resources: Use online resources, such as factoring calculators and worksheets, to practice factoring polynomials.