Find all the real zeros of the polynomial f(x)= X^4-80x^2+1024 and determine the multiplicity of each.
let
$\displaystyle u=x^2 $
$\displaystyle f(u) = u^2 - 80u +1024 $
$\displaystyle u = \frac{-(-80) \mp \sqrt{(80)^2 -4(1024)}}{2} $
$\displaystyle u = \frac{80 \mp 48}{2} $
$\displaystyle u = 40+24 = 64 $
$\displaystyle u = 40-24 = 16$
so
$\displaystyle f(u) = (u-16)(u-64) $ but
$\displaystyle f(x) = (x^2-16)(x^2-64) $
$\displaystyle f(x)=(x-4)(x+4)(x-8)(x+8) $
roots
$\displaystyle x={-8,-4,4,8}$
the multiplicity is one for all roots