Question

Let a ≠ 0 and p(x) be a polynomial of degree greater than 2. If p(x) leaves remainders a and –a, when divided respectively by (x + a) and (x – a), then the remainder when p(x) is divided by (x² – a²) is 

(a) –2x 

(b) –x 

(c) 0 

(d) 2a

Answer 1

(b) -x

Let rx + t be the remainder, q(x) be the quotient when p(x) is divided by x² – a².

p(x) = (x² – a²). qx + rx + t                  ...(i) 

Given, p(x) leaves remainders a and –a respectively when divided by (x + a) and (x – a).

∴ p(–a) = a and p(a) = – a 

Putting x = – a in (i), we get 

p(–a) = 0. q(–a) + (– ra + t) 

⇒ a = – ra + t                     ...(ii) 

Putting x = a, in (i), we get 

p(a) = 0.q (a) + (ra + t) 

⇒ –a = ra + t                   ...(iii) 

∴ Adding (ii) and (iii), we get 2t = 0 ⇒ t = 0 ⇒ r = –1 

∴ Required remainder = rx + t = – x.