SOLUTION: How many complex roots does the polynomial equation have? 5x^3−4x+1=0 2 3 4 5

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Question 1124403: How many complex roots does the polynomial equation have?
5x^3−4x+1=0


2
3
4
5
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

a polynomial equation of degree n has exactly n roots
you have a polynomial equation of degree 3 has exactly highlight%283%29 roots
5x%5E3-4x%2B1=0+
here are they:
5x%5E3-4x%2B1=0+...factor; add 5x%5E2-5x%5E2, and write -4x as x-5x
5x%5E3%2B5x%5E2-5x%5E2%26%238722%3B5x%2Bx%2B1=0+...group
%285x%5E3%2B5x%5E2%29-%285x%5E2%2B5x%29%2B%28x%2B1%29=0+

5x%5E2%28x%2B1%29-5x%28x%2B1%29%2B%28x%2B1%29=0+

%28x+%2B+1%29+%285x%5E2+-+5x+%2B+1%29+=+0

one real solution is: %28x+%2B+1%29++=+0=>x=-1
and, since your function is 3rd degree, there will be two more solutions

use quadratic formula to solve %285x%5E2+-+5x+%2B+1%29+=+0:

In this case a=5,b=-5,c=1

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-%28-5%29+%2B-+sqrt%28+%28-5%29%5E2-4%2A5%2A1+%29%29%2F%282%2A5%29+
x+=+%285+%2B-+sqrt%28+25-20+%29%29%2F10+
x+=+%285+%2B-+sqrt%28+5+%29%29%2F10+
x+=+%285%2F10+%2B-+sqrt%28+5+%29%2F10%29+
x+=+%281%2F2+%2B-+sqrt%28+5+%29%2F10%29+
solutions:
x+=+1%2F2+%2B+sqrt%28+5+%29%2F10+
x+=+1%2F2+-+sqrt%28+5+%29%2F10+


+graph%28+600%2C+600%2C+-5%2C+5%2C+-5%2C+5%2C+5x%5E3-4x%2B1%29+



Answer by ikleyn(52784) About Me  (Show Source):
You can put this solution on YOUR website!
.
I can answer this question without solving the equation.

The polynomial is of degree 3 - hence, it has 3 (three, THREE) complex roots.

No more and no less - exactly 3.