Question #ba224

1 Answer
冠廷 李.
May 2, 2017

two real roots #x=3and2#
two imaginary roots #x=+-i#

Explanation:

before searching for the zeros in f(x)
we need to know there are two conditions
1. four real roots
2.two real roots & two imaginary roots
imaginary roots always be pair!

graph{y=x^4-5x^3+7x^2-5x+6 [-8.89, 8.89, -4.44, 4.45]}
let #f(x)=x^4-5x^3+7x^2-5x+6#
if x=3 #f(3)=81-135+63-15+6=0#
so #f(x)#can be divided by #(x-3)#

#(x^4-5x^3+7x^2-5x+6)/(x-3)=x^3-2x^2+x-2#

the next step also use by the same method

graph{y=x^3-2x^2+x-2 [-8.89, 8.89, -4.44, 4.45]}
let#g(x)=x^3-2x^2+x-2#
if #x=2#
#g(x)=8-8+2-2=0#

#g(x)/(x-2)=x^2+1#
exist two imaginary roots #+-i# in the equation #x^2+1#

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