Question

f(x) = x⁴ – 2x³ + 3x² – ax + b is a polynomial such that when it is divided by (x – 1) and (x + 1), the remainders are respectively 5 and 19. Determine the remainder when f(x) is divided by (x – 2). 

(a) 6 

(b) 10 

(c) 2 

(d) 8

Answer 1

(b) 10

When f(x) = x⁴ – 2x³ + 3x² – ax + b is divided by (x – 1) and (x + 1), the remainders are 5 and 19 respectively. 

i.e., f(1) = 5 and f(–1) = 19 

⇒ 1 – 2 + 3 – a + b = 5 and 1 + 2 + 3 + a + b = 19 

⇒ –a + b = 3 and a + b = 13

Adding the two equations, we get 2b = 16 ⇒ b = 8 ⇒ a = 5

∴ f(x) = x⁴ – 2x³ + 3x² – ax + b 

= x⁴ – 2x³ + 3x² – 5x + 8 

∴ Remainder, when f(x) is divided by (x – 2) is equal to f(2) 

∴ f(2) = 2⁴ – 2.2³ + 3.2² – 5.2 + 8 

= 16 – 16 + 12 – 10 + 8 = 10.