Question

Polynomials bx² + x + 5 and bx³ – 2x + 5 are divided by polynomial x – 3 and the remainders are m and n respectively. If m – n = 0, then find the value of b.

Answer 1

When polynomial bx² + x + 5 is divided by (x – 3), the remainder is m.

 ∴ By remainder theorem, 

Remainder = p(3) = m 

p(x) = bx² + x + 5 

∴ p(3) = b(3)² + 3 + 5 

∴m = b(9) + 8

m = 9b + 8 …(i) 

When polynomial bx³ – 2x + 5 is divided by x – 3 the remainder is n 

∴ remainder = p(3) = n 

p(x) = bx³ – 2x + 5 

∴ P(3)= b(3)³ – 2(3) + 5 

∴ n = b(27) – 6 + 5 

∴ n = 27b – 1 …(ii)

 Now, m – n = 0 …[Given] 

∴ m = n

∴ 9b + 8 = 27b – 1 …[From (i) and (ii)] 

∴ 8 + 1 = 27b – 9b 

∴ 9 = 18b 

∴ b =\(\frac1{2}\)