Question

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.

Answer 1

Let p(x) = 2x³ + ax2 + bx - 14

Given, (x – 2) is a factor of p(x),

⇒ Remainder = p(2) = 0

⇒ 2(2)³ + a(2)² + b(2) – 14 = 0

⇒ 16 + 4a + 2b – 14 = 0

⇒ 4a + 2b + 2 = 0

⇒ 2a + b + 1 = 0 ...(1)

Given, when p(x) is divided by (x – 3), it leaves a remainder 52

∴ p(3) = 52

∴ 2(3)³ + a(3)² + b(3) – 14 = 52

⇒ 54 + 9a + 3b - 14 - 52 = 0

⇒ 9a + 3b – 12 = 0

⇒ 3a + b – 4 = 0 ...(2)

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

a – 5 = 0 ⇒ a = 5

From (1),

10 + b + 1 = 0 ⇒ b = –11