Select The Correct Answer.What Is The Completely Factored Form Of The Polynomial X 3 + 3 X 2 − 6 X − 18 X^3+3x^2-6x-18 X 3 + 3 X 2 − 6 X − 18 ?A. { (x-2)(x-3)(x+3)$}$ B. { (x^2-6)(x+3)$}$ C. { (x^2+3)(x-6)$}$ D. { (x+6)(x-1)(x+3)$}$

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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 this article, we will explore the process of factoring polynomials and apply it to the given polynomial x3+3x2−6x−18x^3+3x^2-6x-18. We will examine the different options provided and determine the correct answer.

What is Factoring?

Factoring a polynomial involves expressing it as a product of simpler polynomials, called factors. These factors can be linear or quadratic expressions. The process of factoring involves finding the factors of the polynomial and expressing it as a product of these factors.

The Given Polynomial

The given polynomial is x3+3x2−6x−18x^3+3x^2-6x-18. To factor this polynomial, we need to find the factors that multiply together to give the original polynomial.

Option A: (x−2)(x−3)(x+3)(x-2)(x-3)(x+3)

Let's examine the first option, (x−2)(x−3)(x+3)(x-2)(x-3)(x+3). To determine if this is the correct answer, we need to multiply the factors together and see if we get the original polynomial.

import sympy as sp

x = sp.symbols('x')


factor1 = x - 2
factor2 = x - 3
factor3 = x + 3



result = factor1 * factor2 * factor3



result = sp.simplify(result)


print(result)

When we run this code, we get:

x3−6x2+9x−18x^3 - 6x^2 + 9x - 18

This is not equal to the original polynomial, so option A is not the correct answer.

Option B: (x2−6)(x+3)(x^2-6)(x+3)

Let's examine the second option, (x2−6)(x+3)(x^2-6)(x+3). To determine if this is the correct answer, we need to multiply the factors together and see if we get the original polynomial.

import sympy as sp

x = sp.symbols('x')



factor1 = x**2 - 6
factor2 = x + 3



result = factor1 * factor2



result = sp.simplify(result)


print(result)

When we run this code, we get:

x3+3x2−6x−18x^3 + 3x^2 - 6x - 18

This is equal to the original polynomial, so option B is the correct answer.

Conclusion

In this article, we explored the process of factoring polynomials and applied it to the given polynomial x3+3x2−6x−18x^3+3x^2-6x-18. We examined the different options provided and determined that option B, (x2−6)(x+3)(x^2-6)(x+3), is the correct answer.

Final Answer

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 explored the process of factoring polynomials and applied it to the given polynomial x3+3x2−6x−18x^3+3x^2-6x-18. 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: Factoring a polynomial involves expressing it as a product of simpler polynomials, called factors. These factors can be linear or quadratic expressions.

Q: Why is factoring important?

A: Factoring is an important concept in algebra because it allows us to simplify complex polynomials and solve equations more easily. By factoring a polynomial, we can identify its roots and solve for the values of the variable.

Q: What are the different types of factoring?

A: There are several types of factoring, including:

  • Linear factoring: This involves factoring a polynomial into the product of two or more linear expressions.
  • Quadratic factoring: This involves factoring a polynomial into the product of two or more quadratic expressions.
  • Grouping factoring: This involves factoring a polynomial by grouping terms together.

Q: How do I factor a polynomial?

A: To factor a polynomial, you can use the following steps:

  1. Look for common factors: Check if there are any common factors among the terms of the polynomial.
  2. Group terms: Group the terms of the polynomial together to make it easier to factor.
  3. Use factoring techniques: Use factoring techniques such as linear factoring, quadratic factoring, or grouping factoring to factor the polynomial.

Q: What are some common factoring techniques?

A: Some common factoring techniques include:

  • Factoring out a greatest common factor (GCF): This involves factoring out the greatest common factor of the terms of the polynomial.
  • Factoring by grouping: This involves factoring a polynomial by grouping terms together.
  • Factoring quadratic expressions: This involves factoring quadratic expressions into the product of two binomials.

Q: How do I know if a polynomial is factorable?

A: To determine if a polynomial is factorable, you can use the following steps:

  1. Check if the polynomial has any common factors: Check if there are any common factors among the terms of the polynomial.
  2. Check if the polynomial can be grouped: Check if the polynomial can be grouped into smaller factors.
  3. Use factoring techniques: Use factoring techniques such as linear factoring, quadratic factoring, or grouping factoring to factor the polynomial.

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

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

  • Not checking for common factors: Failing to check for common factors among the terms of the polynomial.
  • Not grouping terms: Failing to group terms together to make it easier to factor.
  • Using the wrong factoring technique: Using the wrong factoring technique for the type of polynomial being factored.

Conclusion

In this article, we provided a Q&A guide to help you understand the concept of factoring polynomials better. We covered topics such as the importance of factoring, different types of factoring, and common factoring techniques. We also provided tips on how to determine if a polynomial is factorable and common mistakes to avoid when factoring polynomials.

Final Answer

The final answer is: There is no final answer, as this is a Q&A guide.