Lcm of x cubed-x squared+ x-1 and x squared -1
x^3 -x^2 + x -1
Factor by grouping
x^2(x-1) + 1 (x-1)
(x^2+1)(x-1)
For the second one
x^2 -1 factors to (x-1)(x+1)
The LCM (x^2 +1)(x-1)(x+1)
x^3-x^2+x-1 = (x^2+1)(x-1)
x^2-1 = (x+1)(x-1)
So, the LCM is
(x^2+1)(x-1)(x+1)
= (x^2+1)(x^2-1)
= x^4-1
To find the least common multiple (LCM) of two polynomials, we need to factorize them completely and then identify the common factors with their respective highest powers.
Let's factorize the two polynomials:
1. Polynomial 1: x³ - x² + x - 1
Since there is no common factor to factor out, we cannot simplify this polynomial any further.
2. Polynomial 2: x² - 1
This is a difference of squares, which can be factored as:
(x - 1)(x + 1)
Now, let's identify the common factors with their highest powers:
Polynomial 1: x³ - x² + x - 1
Polynomial 2: (x - 1)(x + 1)
To find the LCM, we consider the common factors and their highest powers:
(x - 1) is a common factor with a power of 1.
(x + 1) is a common factor with a power of 1.
x³ - x² + x - 1 is a unique factor.
Now, we multiply the common factors along with their highest powers and the unique factors:
LCM = (x - 1)(x + 1)(x³ - x² + x - 1)
Therefore, the LCM of x³ - x² + x - 1 and x² - 1 is (x - 1)(x + 1)(x³ - x² + x - 1).