JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Mock Test - Complex Numbers and Quadratic Equations

Let \[a\ne 0\]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 \[{{x}^{2}}-{{a}^{2}}\]is

A) 2x

B) -2X

C) x

D) -x

Correct Answer: D

Solution :

[d] We are given that \[p(-a)=a\text{ }and\text{ }p(a)=-a\]

[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

\[p(a)=0+Aa+B\Rightarrow -a=Aa+B\] (2)

And \[p(-a)=0-aA+B\Rightarrow a=-aA+B\] (3)

Solving (2) and (3), we get

B=0 and A=-1

Hence, the required remainder is-x.