Select The Correct Answer.What Is The Completely Factored Form Of This 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 (x-2)(x-3)(x+3 ( X − 2 ) ( X − 3 ) ( X + 3 ]B. ( X 2 − 6 ) ( X + 3 (x^2-6)(x+3 ( X 2 − 6 ) ( X + 3 ]C. ( X 2 + 3 ) ( X − 6 (x^2+3)(x-6 ( X 2 + 3 ) ( X − 6 ]D. ( X + 6 ) ( X − 1 ) ( X + 3 (x+6)(x-1)(x+3 ( 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 also discuss the importance of factoring polynomials and provide tips for factoring different types of polynomials.

What is Factoring?

Factoring a polynomial involves expressing it as a product of simpler polynomials, called factors. Each factor is a polynomial that, when multiplied together, gives the original polynomial. Factoring polynomials is an essential skill in algebra, as it allows us to simplify complex expressions, solve equations, and analyze functions.

Types of Factoring

There are several types of factoring, including:

  • Greatest Common Factor (GCF) Factoring: This involves factoring out the greatest common factor of all 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.
  • Grouping Factoring: This involves grouping the terms in the polynomial into pairs and factoring each pair separately.

Factoring the Given Polynomial

To factor the given polynomial x3+3x2−6x−18x^3 + 3x^2 - 6x - 18, we can start by looking for the greatest common factor of all the terms. In this case, the greatest common factor is 1, so we cannot factor out any common factor.

Next, we can try to factor the polynomial by grouping the terms into pairs. We can group the first two terms together and the last two terms together:

x3+3x2−6x−18=(x3+3x2)−(6x+18)x^3 + 3x^2 - 6x - 18 = (x^3 + 3x^2) - (6x + 18)

Now, we can factor out a common factor from each pair:

(x3+3x2)−(6x+18)=x2(x+3)−6(x+3)(x^3 + 3x^2) - (6x + 18) = x^2(x + 3) - 6(x + 3)

Notice that both pairs have a common factor of (x+3)(x + 3). We can factor this out to get:

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

Therefore, the completely factored form of the polynomial is (x2−6)(x+3)(x^2 - 6)(x + 3).

Conclusion

Factoring polynomials is an essential skill in algebra that involves expressing a polynomial as a product of simpler polynomials. 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 also discussed the importance of factoring polynomials and provided tips for factoring different types of polynomials. By following these tips and practicing factoring, you can become proficient in factoring polynomials and apply this skill to solve equations and analyze functions.

Answer

The completely factored form of the polynomial x3+3x2−6x−18x^3 + 3x^2 - 6x - 18 is:

(x^2 - 6)(x + 3)

This matches option B in the given choices.

Discussion

This problem requires the student to factor a polynomial using the method of grouping. The student must first identify the greatest common factor of all the terms, and then group the terms into pairs. The student must then factor out a common factor from each pair and combine the results to get the completely factored form of the polynomial.

Tips for Factoring

Here are some tips for factoring polynomials:

  • Look for the greatest common factor: The greatest common factor is the largest factor that divides all the terms in the polynomial.
  • Group the terms into pairs: Grouping the terms into pairs can help you identify common factors.
  • Factor out a common factor: Once you have identified a common factor, you can factor it out to simplify the polynomial.
  • Combine the results: Combine the results of factoring out common factors to get the completely factored form of the polynomial.

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 and apply it to different types of polynomials.

Q: What is the greatest common factor (GCF) of a polynomial?

A: The greatest common factor (GCF) of a polynomial is the largest factor that divides all the terms in the polynomial. The GCF can be a constant, a variable, or a combination of both.

Q: How do I factor out the GCF from a polynomial?

A: To factor out the GCF from a polynomial, you need to identify the GCF and then divide each term in the polynomial by the GCF. The result will be a new polynomial with the GCF factored out.

Q: What is the difference of squares formula?

A: The difference of squares formula is a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b). This formula can be used to factor the difference of two squares.

Q: How do I factor the difference of two squares?

A: To factor the difference of two squares, you need to identify the two squares and then apply the difference of squares formula. The result will be a new polynomial with the difference of squares factored.

Q: What is the sum and difference of cubes formula?

A: The sum and difference of cubes formula is a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2) and a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2). These formulas can be used to factor the sum and difference of two cubes.

Q: How do I factor the sum and difference of two cubes?

A: To factor the sum and difference of two cubes, you need to identify the two cubes and then apply the sum and difference of cubes formula. The result will be a new polynomial with the sum and difference of cubes factored.

Q: What is grouping factoring?

A: Grouping factoring is a method of factoring polynomials by grouping the terms into pairs and then factoring out a common factor from each pair.

Q: How do I factor a polynomial using grouping?

A: To factor a polynomial using grouping, you need to group the terms into pairs and then identify a common factor that can be factored out from each pair. The result will be a new polynomial with the common factor factored out.

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

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

  • Not identifying the GCF: Failing to identify the GCF can lead to incorrect factoring.
  • Not applying the difference of squares formula: Failing to apply the difference of squares formula can lead to incorrect factoring.
  • Not applying the sum and difference of cubes formula: Failing to apply the sum and difference of cubes formula can lead to incorrect factoring.
  • Not grouping the terms correctly: Failing to group the terms correctly can lead to incorrect factoring.

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 understand the concept of factoring polynomials and apply it to different types of polynomials. By following the tips and avoiding common mistakes, you can become proficient in factoring polynomials and apply this skill to solve equations and analyze functions.

Additional Resources

For more information on factoring polynomials, you can refer to the following resources:

  • Algebra textbooks: Many algebra textbooks provide detailed explanations and examples of factoring polynomials.
  • Online resources: Websites such as Khan Academy, Mathway, and Wolfram Alpha provide interactive lessons and examples of factoring polynomials.
  • Practice problems: Practice problems can help you reinforce your understanding of factoring polynomials and develop your problem-solving skills.

By following these resources and practicing factoring, you can become proficient in factoring polynomials and apply this skill to solve equations and analyze functions.