Final answer:
To determine which polynomial is divisible by (x - 1), substitute (x - 1) into each polynomial and check if the result is 0. Only polynomial B satisfies this condition.
Step-by-step explanation:
To determine which polynomial is divisible by (x - 1), we can use the remainder theorem. According to the theorem, if we substitute the divisor, (x - 1), into the polynomial and the result is 0, then the polynomial is divisible by the divisor.
Let's substitute (x - 1) into each polynomial:
A. p(x) = 2x^3 - 3x^2 + 2x + 1 → p(1) = 2(1)^3 - 3(1)^2 + 2(1) + 1 = 2 - 3 + 2 + 1 = 2 ≠ 0
B. p(x) = 5x^3 - 4x^2 - x → p(1) = 5(1)^3 - 4(1)^2 - 1 = 5 - 4 - 1 = 0
C. p(x) = 3x^3 + 2x^2 - x → p(1) = 3(1)^3 + 2(1)^2 - 1 = 3 + 2 - 1 = 4 ≠ 0
D. p(x) = x^3 + 2x^2 + 3x + 2 → p(1) = (1)^3 + 2(1)^2 + 3(1) + 2 = 1 + 2 + 3 + 2 = 8 ≠ 0
Therefore, the polynomial that is divisible by (x - 1) is B. p(x) = 5x^3 - 4x^2 - x.