Question

If $x^4-3x^3+2x^2+x-1$ is divided by $(x-2)$, Then remainder will be

Answer 1

Given that:

$P\left(x\right)=x^4-3x^3+2x^2+x-1$

WE know that if $P\left(x\right)$ be a polynomial then there exist two polynomial $Q\left(x\right)\&R\left(x\right)$such that:

$P\left(x\right)=(x-2)Q\left(x\right)+R\left(x\right)...........\left(1\right)$

Now

Rearrange given polynomial as:

$x^4-3x^3+2x^2+x-1=x^4-2x^3-x^3+2x^2+x-2+1$

$x^4-3x^3+2x^2+x-1=x^3(x-2)-x^2(x-2)+1(x-2)+1$

$x^4-3x^3+2x^2+x-1=(x^3-x^2+1)(x-2)+1$

$P\left(x\right)=(x^3-x^2+10)(x-2)+\mathrm{1..............}\left(2\right)$

Compering equation $1\&2$

$R\left(x\right)=1$

$Remainderis1whenP\left(x\right)isdividedbyx-2$