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, the remainder when p(x) is divided by x2 - a2 is
(A) 2x
(B) - 2x
(C) x
(D) - X
​

Answer 1

Answer:

D) -X

Step-by-step explanation:

We are given that p(−a)=a and p(a)=−a

[When a polynomial f(x) is divided by x−a , remainder is f(a)].

Let the remainder, when p(x) is divided by x

2

−a

2

, be Ax+B. Then,

p=(x)=Q(x)(x

2

−a

2

)+Ax+B (1)

where Q(x) is the quotient. Putting x=a and −a in (1), we get

p(a)=0+Aa+B⇒−a=Aa+B (2)

and p(−a)=0−aA+B⇒a=−aA+B (3)

Solving (2) and (3), we get

B=0 and A=−1

Hence, the required remainder is −x.