If x^2 - x - 1 = 0, what is the value of x^3 - 2x^2?
Divide the polynomial $x^3 -2x^2$ by the polynomial $x^2-x-1$. The remainder is the answer. Do you see why this is correct?
Rewrite \(x^2-x-1=0 \) as
\(x^2=x+1\) (1)
Mutiply (1) by x
\(x^3=x^2+x\) Rewrite as
\(x^3-x^2=x \) (2)
Subtract (1) from (2)
\(x^3-x^2-x^2=x-(x+1)\)
Simplify
\(x^3-2x^2=-1\)
