Question

The remainder when $f\left(x\right)=x^{45}$ is divided by $x^2-1$
Is __________.

Answer 1

Given:$f\left(x\right)=x^{45}$ is divided by $x^2-1$

Let, the quotient be 𝑥 and remainder be 𝑎𝑥+𝑏.

$\therefore x^{45}=q(x^2-1)+(ax+b)\dots \dots \dots \dots \left(1\right)$

Now, put 𝑥=1 in (1), we get

$\Rightarrow 1=q(1-1)+a+b$

$\Rightarrow a+b=1\dots \dots \dots .\left(2\right)$

Now, put 𝑥=−1 in (1), we get

$\Rightarrow -1=q(1-1)-a+b$

$\Rightarrow -a+b=-1\dots \dots \dots .\left(3\right)$

By adding (2) and (3) we get,

$2b=0\Rightarrow b=0$

Put 𝑏=0 in (2) ,we get 𝑎=1
$\therefore$Remainder =𝑥

Hence, the remainder is 𝑥.