Question #ba224

1 Answer
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]}
letg(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