Question

Use the factor theorem to determine that x - 1 is a factor of x⁶ - x⁵ + x⁴ - x³ + x² - x + 1.

Answer 1

Let f(x) = x⁶ - x⁵ + x⁴ - x³ + x² - x + 1 to check whether x - 1 is a factor of x⁶ - x⁵ + x⁴ - x³ + x² - x + 1 we find f(1).
Put x = 1 in equation (i) we get
f(1) = (1)⁶ - (1)⁵ + (1)⁴ - (1)³ + (1)² - (1) + 1
= 1 - 1 + 1 - 1 + 1 - 1 + 1
= 4 - 3
= 1.
Since, f(1) ≠ 0, So by factor theorem (x - 1) is not a factor of f(x).