If (x - 2) is a factor of the expression 2x³ + ax² + bx - 14 and when the expression is divided by (x - 3), it leaves a remainder 52, find the values of a and b.

Let f(x) = 2x³ + ax² + bx - 14

Since, (x − 2) is a factor of f(x) then f(2) = 0.

⇒ 2(2)³ + a(2)² + b(2) - 14 = 0

⇒ 2(8) + 4a + 2b - 14 = 0

⇒ 16 + 4a + 2b - 14 = 0

⇒ 4a + 2b + 2 = 0

⇒ 4a + 2b = -2

⇒ 2(2a + b) = -2

⇒ 2a + b = $\dfrac{-2}{2}$

⇒ 2a + b = -1 ….(1)

On dividing f(x) by (x − 3), the remainder is 52,

By remainder theorem,

⇒ f(3) = 52

⇒ 2(3)³ + a(3)² + b(3) - 14 = 52

⇒ 2(27) + 9a + 3b - 14 = 52

⇒ 54 + 9a + 3b - 14 = 52

⇒ 9a + 3b + 40 = 52

⇒ 9a + 3b = 52 - 40

⇒ 9a + 3b = 12

⇒ 3(3a + b) = 12

⇒ 3a + b = $\dfrac{12}{3}$

⇒ 3a + b = 4 ….(2)

Subtracting equation (1) from (2), we get :

⇒ 3a + b - (2a + b) = 4 - (-1)

⇒ a = 4 + 1

⇒ a = 5.

Substituting a = 5 in equation (1), we get :

⇒ 2(5) + b = -1

⇒ 10 + b = -1

⇒ b = -1 - 10

⇒ b = -11.

Hence, the value of a = 5 and b = -11.