Discussion

Explanation:

Let f(x) = x⁴ - ax³ + bx² + 5x - 6,

and g(x) = x² - 5x + 6
g(x) = (x - 3)(x - 2)

Given, x² - 5x + 6 is a factor of x⁴ - ax³ + bx² + 5x - 6.

It means x² - 5x + 6 is completely divisible by x⁴ - ax³ + bx² + 5x - 6.
∴ x⁴ - ax³ + bx² + 5x - 6 is divisible by (x - 3) as well as (x - 2)

Case 1: x⁴ - ax³ + bx² + 5x - 6 is divisible by (x - 3)
⇒ f(3) = 0
⇒ 3⁴ - a × 3³ + b × 3² + 5 × 3 - 6 = 0
⇒ 27a - 9b = 90
⇒ 3a - b = 10   ...(1)

Case 2: x⁴ - ax³ + bx² + 5x - 6 is divisible by (x - 2)
⇒ f(2) = 0
⇒ 2⁴ - a × 2³ + b × 2² + 5 × 2 - 6 = 0
⇒ 8a - 4b = 20
⇒ 2a - b = 5   ...(2)

Solving (1) and (2), we get
a = b = 5

⇒ a - b = 0

Hence, option (b).

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