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 f(x) = 2x³ + ax² + bx - 14   ...(1)
as (x - 2) is factor of (1)
Put x - 2 = 0
⇒ x = 2 in (1)
f(2) = 2(2)³ + a(2)² + b(2) - 14
0 = 16 + 4a + 2b - 14
or 
4a + 2b = -2
or 2a + b = -1    ...(2)
Again when f(x) is divided by (x - 3), it leaves remainder 52
Put x - 3 = 0
⇒ x = 3
f(3) = 2(3)³ + a(3)² + b(3) - 14
52 = 54 + 9a + 3b - 14
52 = 9a + 3b + 40
52 - 40 = 9a + 3b
⇒ 12 = 9a + 3b  
or
4 = 3a + b         ...(3)
Solving (2) and (3)
        3a + b = 4
        2a + b = -1
Sub  -    -      +
              a  = 5
Substitute a = 5 in 3a + b = 4
⇒ 3 x 5 + b = 4
15 + b = 4
⇒ b = 4 - 15
b = -11.