Find the least common multiple of x³ – x² + x – 1 and x² – 1 . Write the answer in factored form.
A.(x + 1)²(x – 1)
B.(x + 1)(x – 1)(x² + 1)
C.(x³ – x² + x – 1)(x² – 1)
D.(x + 1)(x – 1)(x² – 1)
To find the least common multiple (LCM) of two polynomials, we need to factorize both polynomials and then identify the factors that are common to both polynomials.
Given polynomials:
x³ – x² + x – 1 = (x + 1)(x - 1)(x - 1)
x² - 1 = (x + 1)(x - 1)
When we identify the common factors between the two polynomials, we see that the least common multiple will be the product of all unique factors and any remaining factors raised to the highest power they appear at.
Therefore, the LCM is: (x + 1)(x - 1)(x - 1), which is equivalent to (x + 1)(x - 1)²
Thus, the answer is:
A. (x + 1)²(x - 1)