Question

What is x^3−x^2−x+1 divided by x^2−2x+1 ?

Answer 1

 

$\displaystyle\bf\\x^3-x^2-x+1 = \\\\=x^2(x - 1) - (x - 1)= \\\\=(x - 1)(x^2 - 1) =\\\\= (x - 1)(x - 1)(x + 1) = \\\\=(x - 1)^2(x + 1) = \\\\(x^2 - 2x + 1)(x + 1)\\\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\(x^2 - 2x + 1)(x + 1)~\vdots~(x^2 - 2x + 1)\\\\\implies~~x^3-x^2-x+1~~\text{ is divisible by }~~ x^2-2x+1 \\\\Verify:\\\\\frac{x^3-x^2-x+1}{x^2-2x+1}=\frac{(x^2 - 2x + 1)(x + 1)}{x^2-2x+1}=x+1$

Answer 2

X^3-x^2-x+1 divided by x^2-2x+1 equals x + 1