If $x^3 = 2x+1$, what is $x^k$ as a quadratic?

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Suppose that \(x\) satisfies the equation \[x^3=2x+1.\]

1. Show that \[x^4=x+2x^2\qquad\text{and}\qquad x^5=2+4x+x^2.\]

2. For every integer \(k\geq0\), we can uniquely write \[x^k=A_k+B_kx+C_kx^2\] where \(A_k\), \(B_k\), \(C_k\) are integers. So, in part (i), it was shown that \[A_4=0,\ B_4=1,\ C_4=2 \qquad\text{and}\qquad A_5=2,\ B_5=4,\ C_5=1.\] Show that \[A_{k+1}=C_k,\qquad B_{k+1}=A_k+2C_k,\qquad C_{k+1}=B_k.\]

3. Let \[D_k=A_k+C_k-B_k.\] Show that \(D_{k+1}=-D_k\) and hence that \[A_k+C_k=B_k+(-1)^k.\]

4. Let \(F_k=A_{k+1}+C_{k+1}\). Show that \[F_k+F_{k+1}=F_{k+2}.\]