Oxford MAT test, Q3, please help:
Suppose that the equation: $x^4 + Ax^2 + B = (x^2+ax+b)(x^2-ax+b)$
holds for all values of $x$
i. Find $A$ and $B$ in terms of $a$ and $b$.
ii. Use this information to find a factorization of the expression:
$x^4-20x^2+16$
as a product of two quadratics in $x$.
iii. Show that the four solutions of the equation:
$x^4-20x^2+16=0$
can be written as $\pm\sqrt7$ $\pm\sqrt3$
You need to multiply out the quadratics on the RHS $((x^2+ax+b)(x^2-ax+b))$ and equate the corresponding coefficients:
i.e. equate the coefficients of $x^2$ to find an expression for $A$ in terms of $a$ and $b$
Similarly, equate the coefficients of $x^0$ to find an expression for B in terms of $a$ and $b$ (from the equation, $B= b^2$). The rest of the question should be straightforward from there.