Question

Wiedzac, ze x^2-x-1=0 wyznacz wartosc x^3-2x+1

Answer 1

Wykonam dzielenie wielomianow

(x³-2x+1):(x²-x-1)=x+1

-x³+x²+x

-----------

     x²-x+1

    -x²+x+1

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Reszta    2

 

Answer 2

x²-x-1 = 0,    x³-2x+1 = ?

 

x²-x-1 = 0

Δ = 1+4 = 5

√Δ = √5 

x1 = (1-√5)/2

x2 = (1+√5)/2

 

dla x1 = (1-√5)/2

x³-2x+1 = [(1-√5)/2]³-2[(1-√5)/2] +1 = ⅛(1-√5)³ -(1-√5)+1 = ⅛(1-3√5+15-5√5)-1+√5+1 =

             = ⅛(16-8√5)+√5 = 2-√5+√5 = 2

 

dla x2 = (1+√5)/2

x³-2x+1 = [(1+√5)/2]³-2[(1+√5)/2]+1 = ⅛(1+√5)³-(1+√5)+1 = ⅛(1+3√5+15+5√5)-1-√5+1 =

             = ⅛(16+8√5)-1-√5+1 = 2+√5-√5 = 2

 

Odp. x³-2x+1 = 2