Putnam 1938 B1

8. Take either (i) or (ii). (i) Let $A_{i k}$ be the cofactor of $a_{i k}$ in the determinant $$\boldsymbol{d}=\left|\begin{array}{llll} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{array}\right|$$

Let $D$ be the corresponding determinant with $a_{i k}$ replaced by $A_{i k}$. Prove $D=$ $d^{3}$. (page 86) (ii) Let $P(y)=A y^{2}+B y+C$ be a quadratic polynomial in $y$. If the roots of the quadratic equation $P(y)-y=0$ are $a$ and $b(a \neq b)$, show that $a$ and $b$ are roots of the biquadratic equation $P[P(y)]-y=0$. Hence write down a quadratic equation which will give the other two roots, $c$ and $d$, of the biquadratic. Apply this result to solving the following biquadratic equation: $$\left(y^{2}-3 y+2\right)^{2}-3\left(y^{2}-3 y+2\right)+2-y=0$$