1. ## REal Zeros

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

2. Originally Posted by refresh
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

3. Originally Posted by Amer
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
OMG!!! Thank you so much. i have been waiting for help on this for 2 hours you are the best! i was stuck because i didnt know what multiplicity was. so i got confuse ass to is the 2 multipicity or 1 because its a positive and a negative. well thank you so much.

4. Originally Posted by refresh
OMG!!! Thank you so much. i have been waiting for help on this for 2 hours you are the best! i was stuck because i didnt know what multiplicity was. so i got confuse ass to is the 2 multipicity or 1 because its a positive and a negative. well thank you so much.
multiplicity is the power of the roots

example suppose we have this function

$\displaystyle f(x)= (x-3)^2 (x+2)^4(x-1)$

here the root -2 has the multiplicity 4
the root 3 has the multiplicity 2
the root 1 has the multiplicity 1