Question 89
Let $f(x) = x^{4} - x^{3} + 8a x^{2} - a x + a^{2}$ and $g(y) = y^{2} - y + 6a$.
a) Prove that $f(x) = \left(x^{2} - y_{1} x + a\right)\left(x^{2} - y_{2} x + a\right)$, where $y_{1}$ and $y_{2}$ are the roots of the equation $g(y) = 0$.
b) Find all values of $a$ such that the equation $f(x) = 0$ has four distinct positive roots.
a) Prove that $f(x) = \left(x^{2} - y_{1} x + a\right)\left(x^{2} - y_{2} x + a\right)$, where $y_{1}$ and $y_{2}$ are the roots of the equation $g(y) = 0$.
b) Find all values of $a$ such that the equation $f(x) = 0$ has four distinct positive roots.