Divisibility And Factors

Given: The polynomial is:
f(x) = x³ - 4x² + ax + 8
and it is given that (x - 2) is a factor of the polynomial.

Step 1: Recall the Factor Theorem
Factor Theorem states that if (x - c) is a factor of a polynomial f(x), then f(c) = 0.

Step 2: Apply the Factor Theorem
Since (x - 2) is a factor, we substitute x = 2 into the polynomial and set the result equal to zero:

f(2) = (2)³ - 4(2)² + a(2) + 8
= 8 - 16 + 2a + 8

Step 3: Simplify the expression

Combine like terms: f(2) = (8 - 16 + 8) + 2a = 0 + 2a

Step 4: Solve for 'a'

2a = 0
⇒ a = 0

Final Answer: a = 0
Hence, the correct option is Option (1): 0