Find all the real zeros of the polynomial f(x)= X^4-80x^2+1024 and determine the multiplicity of each.

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- Nov 12th 2009, 08:30 PMrefreshREal Zeros
Find all the real zeros of the polynomial f(x)= X^4-80x^2+1024 and determine the multiplicity of each.

- Nov 12th 2009, 09:28 PMAmer
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 - Nov 12th 2009, 09:44 PMrefresh
- Nov 12th 2009, 09:49 PMAmer