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If x^2 - x - 1 = 0, what is the value of x^3 - 2x^2?

 Jun 20, 2021
 #1
avatar+287 
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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?
 

 Jun 20, 2021
 #2
avatar+32668 
+1

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

\(x^3-2x^2=-1\)

 Jun 20, 2021

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