Which of the following shows the factorization of $x^3-x^2+2 x-2$?
A) $\left(x^2+2\right)(x-1)$
B) $(x+2)(x-1)$
C) $(x+2)(x+1)$
D) $\left(x^2+2\right)(x=1)$
Answer (1)
Group the terms: ( x 3 − x 2 ) + ( 2 x − 2 ) .
Factor out common factors: x 2 ( x − 1 ) + 2 ( x − 1 ) .
Factor out the common binomial: ( x 2 + 2 ) ( x − 1 ) .
The factorization is ( x 2 + 2 ) ( x − 1 ) .
Explanation
Understanding the Problem We are given the polynomial x 3 − x 2 + 2 x − 2 and asked to find its factorization from the given options.
Grouping Terms We can use factoring by grouping to factorize the polynomial. First, we group the first two terms and the last two terms: ( x 3 − x 2 ) + ( 2 x − 2 ) .
Factoring out Common Factors Next, we factor out the greatest common factor from each group. From the first group, we can factor out x 2 , and from the second group, we can factor out 2 : x 2 ( x − 1 ) + 2 ( x − 1 ) .
Factoring out the Common Binomial Now, we can factor out the common binomial factor ( x − 1 ) from the entire expression: ( x 2 + 2 ) ( x − 1 ) .
Identifying the Correct Option Comparing our result with the given options, we see that option A, ( x 2 + 2 ) ( x − 1 ) , matches our factorization.
Final Answer Therefore, the correct factorization of x 3 − x 2 + 2 x − 2 is ( x 2 + 2 ) ( x − 1 ) .
Examples
Factoring polynomials is a fundamental skill in algebra and is used in various applications, such as solving equations, simplifying expressions, and analyzing functions. For example, in physics, you might encounter a polynomial expression representing the trajectory of a projectile. Factoring this polynomial can help you determine the points at which the projectile hits the ground or reaches its maximum height. Similarly, in engineering, factoring polynomials can be used to analyze the stability of a system or to design a filter with specific characteristics. By understanding how to factor polynomials, you can gain valuable insights into the behavior of these systems and make informed decisions.
