Question

(x – 2) is a factor of the expression x³ + ax² + bx + 6. When this expression is divided by (x – 3), it leaves the remainder 3. Find the values of a and b. 

Answer 1

As x – 2 is a factor of
f(x) = x³ + ax² + bx + 6
∴ f(2) = 0
∴ (2)³ + a(2)² + b(2) + 6 = 0
⇒ 8 + 4a + 2b + 6 = 0
⇒ 4a + 2b = –14
⇒ 2a + b = –7                 ....(i)
as on dividing f(x) by x – 3
remainder = 3
∴ f(3) = 3
∴ (3)³ + a(3)² + b(3) + 6 = 3
⇒ 27 + 9a + 3b + 6 = 3
⇒ 9a + 3b = –30
⇒ 3a + b = –10                ....(ii)
Solving simultaneously equation (i) and (ii),
∴                 2a + b = –7
                   3a + b – 10
Subtracting  –      –    +
                       –a = 3       
                         a = –3     
Subtracting value of a in equation (i)
2(–3) + b = –7
∴ –6 + b = –7
∴ b = –1
∴ a = –3, b = –1.