Select The Correct Answer.One Factor Of The Polynomial 2 X 3 − 3 X 2 − 3 X + 2 2x^3 - 3x^2 - 3x + 2 2 X 3 − 3 X 2 − 3 X + 2 Is ( X − 2 (x-2 ( X − 2 ]. Which Expression Represents The Other Factor, Or Factors, Of The Polynomial?A. ( 2 X + 1 ) ( X − 1 (2x+1)(x-1 ( 2 X + 1 ) ( X − 1 ]B. ( 2 X 2 + 1 (2x^2+1 ( 2 X 2 + 1 ]C.

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Introduction

Polynomial factorization is a fundamental concept in algebra that involves breaking down a polynomial into simpler factors. In this article, we will explore the process of factorizing a given polynomial and identify the correct factors. We will use the polynomial 2x3−3x2−3x+22x^3 - 3x^2 - 3x + 2 as an example and determine the other factor or factors of the polynomial.

Understanding Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of simpler polynomials, called factors. Each factor is a polynomial that, when multiplied together, results in the original polynomial. The process of factorization involves identifying the common factors of the polynomial and expressing them as separate factors.

The Given Polynomial

The given polynomial is 2x3−3x2−3x+22x^3 - 3x^2 - 3x + 2. We are told that one factor of the polynomial is (x−2)(x-2). Our goal is to determine the other factor or factors of the polynomial.

Step 1: Factor Out the Common Factor

To factor out the common factor, we need to identify the greatest common factor (GCF) of the polynomial. In this case, the GCF is 1, since there is no common factor among the terms. However, we can factor out a common factor of x−2x-2 from the polynomial.

Step 2: Perform Polynomial Division

To perform polynomial division, we divide the polynomial by the known factor (x−2)(x-2). This will give us the other factor or factors of the polynomial.

Step 3: Determine the Other Factor or Factors

After performing polynomial division, we get:

2x3−3x2−3x+2x−2=2x2+x−1\frac{2x^3 - 3x^2 - 3x + 2}{x-2} = 2x^2 + x - 1

This means that the other factor or factors of the polynomial are 2x2+x−12x^2 + x - 1.

Conclusion

In conclusion, we have successfully factorized the polynomial 2x3−3x2−3x+22x^3 - 3x^2 - 3x + 2 and identified the other factor or factors of the polynomial. The correct answer is 2x2+x−12x^2 + x - 1.

Answer Options

Let's compare our answer with the given options:

A. (2x+1)(x−1)(2x+1)(x-1) B. (2x2+1)(2x^2+1) C. (2x2+x−1)(2x^2+x-1)

Our answer matches option C, which is (2x2+x−1)(2x^2+x-1).

Final Answer

The final answer is:

  • C. (2x2+x−1)(2x^2+x-1)

This is the correct answer, as it represents the other factor or factors of the polynomial 2x3−3x2−3x+22x^3 - 3x^2 - 3x + 2.

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Introduction

Polynomial factorization is a fundamental concept in algebra that involves breaking down a polynomial into simpler factors. In our previous article, we explored the process of factorizing a given polynomial and identified the correct factors. In this article, we will provide a Q&A guide to help you understand polynomial factorization and its applications.

Q&A: Polynomial Factorization

Q1: What is polynomial factorization?

A1: Polynomial factorization is the process of expressing a polynomial as a product of simpler polynomials, called factors. Each factor is a polynomial that, when multiplied together, results in the original polynomial.

Q2: Why is polynomial factorization important?

A2: Polynomial factorization is important because it helps us to simplify complex polynomials and solve equations. It also helps us to identify the roots of a polynomial, which is essential in many applications, such as physics, engineering, and economics.

Q3: How do I factor a polynomial?

A3: To factor a polynomial, you need to identify the greatest common factor (GCF) of the polynomial and express it as a separate factor. Then, you need to perform polynomial division to divide the polynomial by the known factor.

Q4: What is the difference between factoring and simplifying a polynomial?

A4: Factoring a polynomial involves expressing it as a product of simpler polynomials, while simplifying a polynomial involves combining like terms to reduce its complexity.

Q5: Can I factor a polynomial with a negative leading coefficient?

A5: Yes, you can factor a polynomial with a negative leading coefficient. In fact, the process of factorization remains the same, regardless of the sign of the leading coefficient.

Q6: How do I determine the number of factors of a polynomial?

A6: To determine the number of factors of a polynomial, you need to perform polynomial division and count the number of factors obtained.

Q7: Can I factor a polynomial with a variable in the denominator?

A7: No, you cannot factor a polynomial with a variable in the denominator. In such cases, you need to rationalize the denominator before attempting to factor the polynomial.

Q8: How do I factor a polynomial with a quadratic factor?

A8: To factor a polynomial with a quadratic factor, you need to identify the quadratic factor and express it as a product of two binomials.

Q9: Can I factor a polynomial with a complex factor?

A9: Yes, you can factor a polynomial with a complex factor. In fact, complex factors can be expressed as a product of two binomials with complex coefficients.

Q10: How do I check if a polynomial is factorable?

A10: To check if a polynomial is factorable, you need to perform polynomial division and see if the remainder is zero. If the remainder is zero, then the polynomial is factorable.

Conclusion

In conclusion, polynomial factorization is a powerful tool in algebra that helps us to simplify complex polynomials and solve equations. By understanding the process of factorization and its applications, you can tackle a wide range of problems in mathematics and other fields.

Final Tips

  • Always start by identifying the greatest common factor (GCF) of the polynomial.
  • Perform polynomial division to divide the polynomial by the known factor.
  • Count the number of factors obtained to determine the number of factors of the polynomial.
  • Rationalize the denominator before attempting to factor a polynomial with a variable in the denominator.
  • Express a quadratic factor as a product of two binomials.
  • Check if a polynomial is factorable by performing polynomial division and seeing if the remainder is zero.