Final answer:
To determine how many times (x - 1) is a factor of the polynomial function g(x) = x^4 - x^3 - 3x^2 + 5x - 27, we divide the polynomial function by (x - 1) using synthetic division. The remainder after the division is -28, indicating that (x - 1) is not a factor of g(x).
Step-by-step explanation:
To determine how many times (x - 1) is a factor of the polynomial function g(x) = x^4 - x^3 - 3x^2 + 5x - 27, we need to divide the polynomial function by (x - 1) using synthetic division:
1 -1 -3 5 -27
----------------------------------------
(x - 1) | 1 0 -3 2 -25
-1
----------
1 -1 -4 3 -28
-1 1 3
-----------
1 0 -1 4 -25
1 -1 -3
-----------
1 1 -2 1 -28
The remainder after the division is -28. If (x - 1) is a factor of g(x), then the remainder should be zero. Since the remainder is not zero, (x - 1) is not a factor of the polynomial function g(x).