Select The Correct Answer.What Are The Factors Of The Polynomial Function G ( X ) = X 3 + 2 X 2 − X − 2 G(x) = X^3 + 2x^2 - X - 2 G ( X ) = X 3 + 2 X 2 − X − 2 ?A. { (x-1)(x+1)$}$ B. { (x-1)(x+1)(x+2)$}$ C. { (x-2)(x-1)(x+2)$}$ D. { (x-2)(x-1)(x+1)$}$

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Introduction

Factoring polynomial functions is a crucial concept in algebra that helps us simplify complex expressions and solve equations. In this article, we will focus on factoring the polynomial function g(x)=x3+2x2−x−2g(x) = x^3 + 2x^2 - x - 2. We will explore the different methods of factoring and identify the correct factors among the given options.

What are Factors?

Before we dive into factoring the polynomial function, let's understand what factors are. Factors are the numbers or expressions that are multiplied together to get another number or expression. In the context of polynomial functions, factors are the expressions that are multiplied together to get the original polynomial.

Factoring Methods

There are several methods of factoring polynomial functions, including:

  • Greatest Common Factor (GCF): This method involves finding the greatest common factor of all the terms in the polynomial.
  • Grouping: This method involves grouping the terms in the polynomial into pairs or groups and factoring out the common factors.
  • Synthetic Division: This method involves using synthetic division to find the factors of the polynomial.
  • Factoring by Grouping: This method involves factoring the polynomial by grouping the terms into pairs or groups and factoring out the common factors.

Factoring the Polynomial Function g(x)=x3+2x2−x−2g(x) = x^3 + 2x^2 - x - 2

To factor the polynomial function g(x)=x3+2x2−x−2g(x) = x^3 + 2x^2 - x - 2, we can use the method of factoring by grouping. This method involves grouping the terms in the polynomial into pairs or groups and factoring out the common factors.

Step 1: Group the Terms

Let's group the terms in the polynomial into pairs or groups:

  • (x3+2x2)(x^3 + 2x^2)
  • (−x−2)(-x - 2)

Step 2: Factor Out the Common Factors

Now, let's factor out the common factors from each group:

  • (x2(x+2))(x^2(x + 2))
  • (−1(x+2))(-1(x + 2))

Step 3: Factor Out the Common Binomial Factor

Now, let's factor out the common binomial factor (x+2)(x + 2) from each group:

  • (x2(x+2))=(x+2)(x2)(x^2(x + 2)) = (x + 2)(x^2)
  • (−1(x+2))=−(x+2)(-1(x + 2)) = -(x + 2)

Step 4: Write the Factored Form

Now, let's write the factored form of the polynomial function:

g(x)=(x+2)(x2−1)g(x) = (x + 2)(x^2 - 1)

Step 5: Factor the Quadratic Expression

Finally, let's factor the quadratic expression (x2−1)(x^2 - 1):

(x2−1)=(x−1)(x+1)(x^2 - 1) = (x - 1)(x + 1)

Step 6: Write the Final Factored Form

Now, let's write the final factored form of the polynomial function:

g(x)=(x+2)(x−1)(x+1)g(x) = (x + 2)(x - 1)(x + 1)

Conclusion

In this article, we factored the polynomial function g(x)=x3+2x2−x−2g(x) = x^3 + 2x^2 - x - 2 using the method of factoring by grouping. We identified the correct factors among the given options and wrote the final factored form of the polynomial function.

Answer

The correct answer is:

  • B. {(x-1)(x+1)(x+2)$}$

Introduction

Factoring polynomial functions is a crucial concept in algebra that helps us simplify complex expressions and solve equations. In our previous article, we explored the different methods of factoring and identified the correct factors among the given options. In this article, we will provide a Q&A guide to help you understand the concept of factoring polynomial functions better.

Q&A

Q: What is factoring in algebra?

A: Factoring in algebra involves expressing a polynomial as a product of simpler polynomials, called factors. This helps us simplify complex expressions and solve equations.

Q: What are the different methods of factoring?

A: There are several methods of factoring polynomial functions, including:

  • Greatest Common Factor (GCF): This method involves finding the greatest common factor of all the terms in the polynomial.
  • Grouping: This method involves grouping the terms in the polynomial into pairs or groups and factoring out the common factors.
  • Synthetic Division: This method involves using synthetic division to find the factors of the polynomial.
  • Factoring by Grouping: This method involves factoring the polynomial by grouping the terms into pairs or groups and factoring out the common factors.

Q: How do I factor a polynomial using the GCF method?

A: To factor a polynomial using the GCF method, follow these steps:

  1. Identify the greatest common factor of all the terms in the polynomial.
  2. Factor out the greatest common factor from each term.
  3. Write the factored form of the polynomial.

Q: How do I factor a polynomial using the grouping method?

A: To factor a polynomial using the grouping method, follow these steps:

  1. Group the terms in the polynomial into pairs or groups.
  2. Factor out the common factors from each group.
  3. Write the factored form of the polynomial.

Q: How do I factor a polynomial using synthetic division?

A: To factor a polynomial using synthetic division, follow these steps:

  1. Choose a root of the polynomial.
  2. Use synthetic division to divide the polynomial by the chosen root.
  3. Write the factored form of the polynomial.

Q: What is the difference between factoring and simplifying?

A: Factoring involves expressing a polynomial as a product of simpler polynomials, while simplifying involves combining like terms to reduce the complexity of the polynomial.

Q: Can I factor a polynomial with a negative coefficient?

A: Yes, you can factor a polynomial with a negative coefficient. Simply factor out the negative sign along with the other factors.

Q: Can I factor a polynomial with a fractional coefficient?

A: Yes, you can factor a polynomial with a fractional coefficient. Simply factor out the fractional coefficient along with the other factors.

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

A: A polynomial is factorable if it can be expressed as a product of simpler polynomials. You can use the methods of factoring to determine if a polynomial is factorable.

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

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

  • Not identifying the greatest common factor.
  • Not grouping the terms correctly.
  • Not factoring out the common factors correctly.
  • Not writing the factored form of the polynomial correctly.

Conclusion

In this article, we provided a Q&A guide to help you understand the concept of factoring polynomial functions better. We covered the different methods of factoring, including the GCF method, grouping method, synthetic division method, and factoring by grouping method. We also discussed common mistakes to avoid when factoring polynomials.

Tips and Tricks

  • Always identify the greatest common factor before factoring a polynomial.
  • Group the terms in the polynomial into pairs or groups to make factoring easier.
  • Use synthetic division to find the factors of a polynomial.
  • Factor out the common factors correctly to avoid mistakes.
  • Write the factored form of the polynomial correctly to avoid mistakes.

Practice Problems

  • Factor the polynomial x2+5x+6x^2 + 5x + 6 using the GCF method.
  • Factor the polynomial x2−7x+12x^2 - 7x + 12 using the grouping method.
  • Factor the polynomial x3+2x2−x−2x^3 + 2x^2 - x - 2 using synthetic division.
  • Factor the polynomial x2+3x−4x^2 + 3x - 4 using factoring by grouping.

Answer Key

  • x2+5x+6=(x+3)(x+2)x^2 + 5x + 6 = (x + 3)(x + 2)
  • x2−7x+12=(x−3)(x−4)x^2 - 7x + 12 = (x - 3)(x - 4)
  • x3+2x2−x−2=(x+2)(x−1)(x+1)x^3 + 2x^2 - x - 2 = (x + 2)(x - 1)(x + 1)
  • x2+3x−4=(x+4)(x−1)x^2 + 3x - 4 = (x + 4)(x - 1)