When a polynomial 2x^3+3x^2+ax+b is divided by (x-2) leaves remainder 2, and (x+2) leaves remainder -2.Find a and b.

p(x) = 2x³ + 3x² + ax + b when divided by x - 2 and x + 2 leaves the remainders 2 , - 2.

Value of a and b

✯ g(x) = x - 2

⟹ 0 = x - 2 [ g(x) = 0 ]

⟹ x = 2

★ p(2) = 2(2)³ + 3(2)² + a(2) + b

⟹ - 2 = 2 * 8 + 3 * 4 + 2a + b

⟹ - 2 - 16 - 12 - 2a = b

⟹ - 30 - 2a = b -- equation (1)

Similarly,

✯ g(x) = x + 2

⟹ 0 = x + 2

⟹ x = - 2

★ p( - 2) = 2( - 2)³ + 3( - 2)² + a ( - 2) + b

Substitute the value of b from equation (1).

⟹ 2 = 2 * ( - 8) + 3 * 4 - 2a - 30 - 2a

⟹ 2a + 2a = - 16 + 12 - 2 - 30

⟹ 4a = - 24

⟹ a = - 24/4

⟹ a = - 6

Substitute the value of a in equation (1).

⟹ - 30 - 2( - 6) = b

⟹ - 30 + 12 = b

⟹ - 18 = b

Therefore,

a = - 6

b = - 18.

I hope it will help you.

Regards.