Review question

# If $x^3 = 2x+1$, what is $x^k$ as a quadratic? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8586

## Question

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 $F_k+F_{k+1}=F_{k+2}.$