Let \(t\) be a root of \(f(x) = x^3 - x + 2\). Evaluate \(t^6 - t^2 + 4t\).
+37146 The only root of x^3-x+2 is -1.5214 (by graphing)
-1.5214^6 - ( -1.5214)^2 +4(-1.5214) = 4.00
+26387 Let t be a root of
\(f(x) = x^3 - x + 2\).
Evaluate
\(t^6 - t^2 + 4t\).
1.
\(\begin{array}{|rcll|} \hline f(t) = t^3 - t + 2 &=& 0 \\ t^3 - t + 2 &=& 0 \\ \mathbf{t^3} &\mathbf{=}&\mathbf{t-2} \\ \hline \end{array}\)
2.
\(\begin{array}{|rcll|} \hline t^6 - t^2 + 4t &=& (t^3)^2 - t^2 + 4t \quad | \quad \mathbf{t^3=t-2} \\ &=& (t-2)^2 - t^2 + 4t \\ &=& t^2-4t + 4 - t^2 + 4t \\ \mathbf{t^6 - t^2 + 4t} &\mathbf{=}&\mathbf{4} \\ \hline \end{array} \)
