Let a≠0 and P(x) be a polynomial of degree greater than 2. If P(x) leaves remainder a and –a when divided respectively by x + a and x – a then  the   remainder when P(x) is divided by ${\mathrm{x}}^2– {\mathrm{a}}^2$ is

$\begin{array}{l}IfP\left(x\right)dividedbyx+aandx-a\\ Then\mathrm{Re}maindersare\\ P(-a)=aandP\left(a\right)=-a\\ Letthe\mathrm{Re}mainderwhenP\left(x\right)isdividedby\\ x^2-a^{2}beAx+B\\ \therefore P\left(x\right)=Q\left(x\right)(x^2-a^2)+(Ax+B)\\ Putx=a\Rightarrow P\left(a\right)=Q\left(a\right)(a^2-a^2)+Aa+B\\ \Rightarrow Aa+B=-a--\left(1\right)\\ Putx=-a\Rightarrow P(-a)=Q(-a)(a^2-a^2)-Aa+B\\ \Rightarrow -Aa+B=a--\left(2\right)\\ \left(1\right)+\left(2\right)\Rightarrow 2B=0\Rightarrow B=0\\ from\left(1\right)Aa=-a\Rightarrow -A=-1\\ \therefore \mathrm{Re}mainder=Ax+B\\ =-x\end{array}$