Factorization Of X³ - X² + 2x - 2 A Step-by-Step Solution
As we delve into the realm of algebra, factorization emerges as a fundamental technique for simplifying expressions and solving equations. In this comprehensive guide, we will embark on a journey to unravel the factorization of the polynomial expression x³ - x² + 2x - 2. Our exploration will involve a step-by-step analysis, empowering you to confidently identify the correct factorization from a set of options. Let's begin by understanding the significance of factorization and its applications in various mathematical contexts.
Understanding Factorization
Factorization, in essence, is the process of expressing a polynomial as a product of simpler expressions, known as factors. These factors, when multiplied together, yield the original polynomial. Factorization serves as a powerful tool in algebra, enabling us to simplify complex expressions, solve equations, and gain deeper insights into the structure of polynomials. In the context of polynomial equations, factorization allows us to identify the roots or solutions, which are the values of the variable that make the equation true. By expressing a polynomial as a product of factors, we can readily determine the roots by setting each factor equal to zero and solving for the variable. This technique is widely used in solving quadratic equations, cubic equations, and higher-degree polynomial equations.
Furthermore, factorization plays a crucial role in simplifying algebraic fractions, which are expressions involving ratios of polynomials. By factoring the numerator and denominator of an algebraic fraction, we can identify common factors that can be canceled, leading to a simplified expression. This simplification is essential in various mathematical operations, such as adding, subtracting, multiplying, and dividing algebraic fractions. Factorization also finds applications in calculus, particularly in the context of integration. When dealing with integrals involving rational functions (ratios of polynomials), factorization techniques are often employed to decompose the integrand into simpler fractions that can be integrated more easily. This process, known as partial fraction decomposition, relies heavily on the ability to factor polynomials.
Exploring the Given Options
Before we embark on the factorization process, let's carefully examine the options presented to us:
A) (x² + 2)(x - 1)
B) (x + 2)(x - 1)
C) (x + 2)(x + 1)
D) (x² + 2)(x = 1)
Our objective is to identify the option that accurately represents the factored form of the polynomial x³ - x² + 2x - 2. To achieve this, we will employ a systematic approach, utilizing the technique of factoring by grouping.
Factoring by Grouping A Step-by-Step Approach
Factoring by grouping is a powerful technique that allows us to factor polynomials with four or more terms by strategically grouping terms and extracting common factors. This method is particularly effective when the polynomial does not have a readily apparent common factor across all terms. Let's apply this technique to our polynomial, x³ - x² + 2x - 2.
- Grouping Terms: The first step involves grouping the terms of the polynomial into pairs. In this case, we can group the first two terms and the last two terms together:
(x³ - x²) + (2x - 2)
- Extracting Common Factors: Next, we identify and extract the greatest common factor (GCF) from each group. From the first group, (x³ - x²), the GCF is x². Factoring out x², we get:
x²(x - 1)
Similarly, from the second group, (2x - 2), the GCF is 2. Factoring out 2, we obtain:
2(x - 1)
- Combining Factors: Now, we have:
x²(x - 1) + 2(x - 1)
Notice that both terms now share a common factor of (x - 1). We can factor out this common factor to obtain:
(x - 1)(x² + 2)
Identifying the Correct Factorization
Comparing our factored expression, (x - 1)(x² + 2), with the options provided, we can clearly see that option A) (x² + 2)(x - 1) matches our result. Therefore, the correct factorization of x³ - x² + 2x - 2 is (x² + 2)(x - 1).
Verifying the Factorization
To ensure the accuracy of our factorization, we can multiply the factors back together and verify that we obtain the original polynomial. Let's multiply (x² + 2) and (x - 1):
(x² + 2)(x - 1) = x²(x - 1) + 2(x - 1) = x³ - x² + 2x - 2
As we can see, the product of the factors matches the original polynomial, confirming that our factorization is correct.
Conclusion
In this comprehensive guide, we have successfully factored the polynomial x³ - x² + 2x - 2 using the technique of factoring by grouping. We identified the correct factorization as (x² + 2)(x - 1), which corresponds to option A). By understanding the principles of factorization and mastering techniques like factoring by grouping, you can confidently tackle a wide range of algebraic problems. Factorization is a fundamental skill in mathematics, with applications spanning various fields, including equation solving, algebraic simplification, and calculus. By honing your factorization abilities, you will gain a valuable tool for navigating the world of mathematics.
This detailed explanation not only provides the correct answer but also delves into the underlying concepts and techniques, empowering you to approach similar problems with confidence. Remember, practice is key to mastering factorization, so continue to explore and challenge yourself with different polynomials. With consistent effort, you will become proficient in factorization and unlock its many applications in mathematics and beyond.
