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

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Jan 12, 2021

#1
+25656
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If

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

$$\begin{array}{|rcll|} \hline \mathbf{x^2 - x - 1} &=& \mathbf{0} \\ x^2 - x &=& 1 \qquad (1) \\ x^2 - 1 &=& x \qquad (2) \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \mathbf{x^3 - 2x + 1} &=& \\ &=& x^2x - x - x + 1 \\ &=& x(x^2-1)-x+1 \quad | \quad \mathbf{x^2-1 = x} \\ &=& x*x-x+1 \\ &=& x^2-x+1 \quad | \quad \mathbf{x^2-x = 1} \\ &=& 1+1 \\ \mathbf{x^3 - 2x + 1}&=& \mathbf{2} \\ \hline \end{array}$$

Jan 12, 2021

#1
+25656
+3

If

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

$$\begin{array}{|rcll|} \hline \mathbf{x^2 - x - 1} &=& \mathbf{0} \\ x^2 - x &=& 1 \qquad (1) \\ x^2 - 1 &=& x \qquad (2) \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \mathbf{x^3 - 2x + 1} &=& \\ &=& x^2x - x - x + 1 \\ &=& x(x^2-1)-x+1 \quad | \quad \mathbf{x^2-1 = x} \\ &=& x*x-x+1 \\ &=& x^2-x+1 \quad | \quad \mathbf{x^2-x = 1} \\ &=& 1+1 \\ \mathbf{x^3 - 2x + 1}&=& \mathbf{2} \\ \hline \end{array}$$

heureka Jan 12, 2021