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

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 determine the correct expression that represents the other factor or factors of the polynomial.

Understanding the Problem

The given polynomial is $2x^3 - 3x^2 - 3x + 2$. We are asked to find the other factor or factors of the polynomial, given that one factor is $(x-2)$. To approach this problem, we need to use the concept of polynomial division and factorization.

Polynomial Division

Polynomial division is a process of dividing a polynomial by another polynomial. In this case, we will divide the given polynomial by the factor $(x-2)$ to find the other factor or factors.

To perform polynomial division, we can use the following steps:

1. Divide the leading term of the polynomial ($2x^3$) by the leading term of the divisor ($x$) to get the first term of the quotient ($2x^2$).
2. Multiply the divisor ($x-2$) by the first term of the quotient ($2x^2$) to get $2x^3 - 4x^2$.
3. Subtract $2x^3 - 4x^2$ from the polynomial ($2x^3 - 3x^2 - 3x + 2$) to get $x^2 - 3x + 2$.
4. Repeat the process by dividing the leading term of the new polynomial ($x^2$) by the leading term of the divisor ($x$) to get the next term of the quotient ($x$).
5. Multiply the divisor ($x-2$) by the next term of the quotient ($x$) to get $x^2 - 2x$.
6. Subtract $x^2 - 2x$ from the new polynomial ($x^2 - 3x + 2$) to get $-x + 2$.
7. Repeat the process by dividing the leading term of the new polynomial ($-x$) by the leading term of the divisor ($x$) to get the next term of the quotient ($-1$).
8. Multiply the divisor ($x-2$) by the next term of the quotient ($-1$) to get $-x + 2$.
9. Subtract $-x + 2$ from the new polynomial ($-x + 2$) to get $0$.

Finding the Other Factor

After performing polynomial division, we find that the other factor or factors of the polynomial are $x^2 - 3x + 2$.

Factoring the Quadratic Expression

The quadratic expression $x^2 - 3x + 2$ can be factored using the following steps:

1. Find two numbers whose product is $2$ and whose sum is $-3$. These numbers are $-2$ and $-1$.
2. Write the quadratic expression as a product of two binomials: $(x-2)(x-1)$.

Conclusion

In conclusion, the other factor or factors of the polynomial $2x^3 - 3x^2 - 3x + 2$ are $(x-2)(x-1)$. This can be verified by multiplying the two factors together to get the original polynomial.

Answer

The correct expression that represents the other factor or factors of the polynomial is:

(x-2)(x-1)

This is option A.

Final Thoughts

Polynomial factorization is a powerful tool in algebra that allows us to break down complex polynomials into simpler factors. By using polynomial division and factoring techniques, we can find the other factor or factors of a polynomial and gain a deeper understanding of the underlying mathematics.

References

• Polynomial Factorization
• Polynomial Division
• Factoring Quadratic Expressions
  Q&A: Polynomial Factorization and Division =============================================

Introduction

In our previous article, we explored the process of polynomial factorization and division. We learned how to break down a polynomial into simpler factors and how to use polynomial division to find the other factor or factors of a polynomial. In this article, we will answer some common questions related to polynomial factorization and division.

Q: What is polynomial factorization?

A: Polynomial factorization is the process of breaking down a polynomial into simpler factors. This can be done using various techniques, including factoring out common factors, using the difference of squares formula, and using polynomial division.

Q: What is polynomial division?

A: Polynomial division is the process of dividing a polynomial by another polynomial. This can be done using long division or synthetic division.

Q: How do I perform polynomial division?

A: To perform polynomial division, follow these steps:

1. Divide the leading term of the polynomial by the leading term of the divisor.
2. Multiply the divisor by the result and subtract the product from the polynomial.
3. Repeat the process with the new polynomial until you get a remainder of 0.

Q: What is the difference between polynomial division and factoring?

A: Polynomial division and factoring are two different techniques used to break down polynomials. Polynomial division involves dividing a polynomial by another polynomial, while factoring involves breaking down a polynomial into simpler factors.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, follow these steps:

1. Find two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
2. Write the quadratic expression as a product of two binomials.

Q: What is the difference of squares formula?

A: The difference of squares formula is a formula used to factor quadratic expressions of the form $a^2 - b^2$. The formula is:

$$a^2 - b^2 = (a+b)(a-b)$$

Q: How do I use the difference of squares formula?

A: To use the difference of squares formula, follow these steps:

1. Identify the quadratic expression as a difference of squares.
2. Apply the formula to factor the expression.

Q: What are some common mistakes to avoid when performing polynomial division?

A: Some common mistakes to avoid when performing polynomial division include:

• Not following the correct order of operations.
• Not multiplying the divisor by the result correctly.
• Not subtracting the product from the polynomial correctly.

Q: How do I check my work when performing polynomial division?

A: To check your work when performing polynomial division, follow these steps:

1. Multiply the divisor by the result and subtract the product from the polynomial.
2. Check that the remainder is 0.
3. Check that the quotient is correct.

Conclusion

In conclusion, polynomial factorization and division are powerful tools in algebra that allow us to break down complex polynomials into simpler factors. By understanding the concepts and techniques involved, we can solve a wide range of problems and gain a deeper understanding of the underlying mathematics.

References

• Polynomial Factorization
• Polynomial Division
• Factoring Quadratic Expressions
• Difference of Squares Formula