Factorise:
(i) [tex]$x^3-2 x^2-x+2$[/tex]

To factor the polynomial x 3 − 2 x 2 − x + 2 , we group and factor by parts, resulting in the expression ( x − 2 ) ( x − 1 ) ( x + 1 ) . This is achieved through finding common factors and recognizing the difference of squares. The complete factorization is confirmed by expanding the factors back to the original polynomial.
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Answered by Anonymous | 2025-07-07

Group the terms: x 3 − 2 x 2 − x + 2 .
Factor out common factors: x 2 ( x − 2 ) − 1 ( x − 2 ) .
Factor out the common binomial: ( x − 2 ) ( x 2 − 1 ) .
Factor the difference of squares: ( x − 2 ) ( x − 1 ) ( x + 1 ) .
The final factorization is ( x − 2 ) ( x − 1 ) ( x + 1 ) ​ .

Explanation

Understanding the Problem We are given the cubic polynomial x 3 − 2 x 2 − x + 2 and our goal is to factorize it completely.

Grouping Terms We will attempt to factorize the polynomial by grouping. This involves grouping the first two terms and the last two terms together.

Factoring out Common Factors From the first group, x 3 − 2 x 2 , we can factor out x 2 , which gives us x 2 ( x − 2 ) . From the second group, − x + 2 , we can factor out − 1 , which gives us − 1 ( x − 2 ) . So, we have: x 3 − 2 x 2 − x + 2 = x 2 ( x − 2 ) − 1 ( x − 2 )

Factoring out the Common Binomial Now, we can see that ( x − 2 ) is a common factor in both terms. We factor out ( x − 2 ) :
x 2 ( x − 2 ) − 1 ( x − 2 ) = ( x − 2 ) ( x 2 − 1 )

Factoring the Difference of Squares We recognize that x 2 − 1 is a difference of squares, which can be factored as ( x − 1 ) ( x + 1 ) . Therefore, we have: ( x − 2 ) ( x 2 − 1 ) = ( x − 2 ) ( x − 1 ) ( x + 1 )

Final Factorization Thus, the complete factorization of the given polynomial is ( x − 2 ) ( x − 1 ) ( x + 1 ) .


Examples
Factoring polynomials is a fundamental skill in algebra and is used in many areas of mathematics and engineering. For example, in physics, you might need to find the roots of a polynomial to determine the equilibrium points of a system. In engineering, factoring can help simplify complex expressions when designing circuits or analyzing structural stability. Let's say you are designing a bridge and need to ensure that the forces acting on it are balanced. The forces might be described by a polynomial equation, and factoring that polynomial can help you find the points where the forces are zero, indicating stable equilibrium.

Answered by GinnyAnswer | 2025-07-07