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

Let p(x) = bx² + x + 5 and q(x) = bx³ - 2x + 5.

The remainder when p(x) = bx² + x + 5 is divided by (x − 3) is m.

By remainder theorem, 
Remainder = p(3) = m

∴ b × (3)² + 3 + 5 = m

⇒ m = 9b + 8        ...(1)

The remainder when q(x) = bx³ − 2x + 5 is divided by (x - 3) is n.

By remainder theorem, 

Remainder = q(3) = n

∴ b × (3)³ − 2 × 3 + 5 = n

⇒ n = 27b − 6 + 5

⇒ n = 27b − 1   ...(2)

Now, 

m − n = 0

⇒ (9b + 8) - (27b − 1) = 0     [Using (1) and (2)]

⇒ 9b − 27b + 8 + 1 = 0

⇒ −18b + 9 = 0

⇒ −18b = −9

`⇒ b = (−9)/(−18) = 1/2`

Thus, the value of b is `1/2`.