I have tried to solve but I can't fnd the roots.

x^5-2x^4+x^3+x^2-2x+1=0

positive roots: 4,2,0

negative roots: 1

imaginary roots: 0,2,4

From rational zeros we get: +/-1 as root of the equation.

So we get,

(x+1)(x-1)(x^3-2x^2+2x-1)  --------------------------(1)

Again applying descartes sign rule on (x^3-2x^2+2x-1) we get:

positive: 3,1

negative: 0

imaginary: 0,2

On applying rational zero test we get: x=1 as positive root.

On long division we get:

(x-1)(x^2-x+1)

Now solving the quadratic equation, (x^2-x+1) we get:

x = (1 +/- sqrt(3)i)/2

So the roots are x =1, -1 , (1 +/- sqrt(3)i)/2
by Level 4 User (9.5k points)
edited
So there are total of two positive roots  +1 with mutiplicity of 2

one negative root -1 with multiplicity of 1

and two imaginary roots with (1 + sqrt(3)i)/2 and (1 - sqrt(3)i)/2

Descarte’s Rule determines how many real roots there are by counting the number of sign changes in the coefficients. I count three.

Rational zeroes indicates that x=±1 could be a root, so put x=1 into the polynomial:

1-2+1+1-2+1=0, so x-1 is a factor.

Use synthetic division to divide by the root:

1 | 1 -2  1 1 -2   1

1  1 -1 0  1  -1

1 -1  0 1 -1 | 0 = x⁴-x³+x-1.

Putting x=1 into this quartic also gives us 0, so x-1 is a factor, making (x-1)² a factor of the original polynomial.

Use synthetic division again:

1 | 1 -1 0 1  -1

1  1 0 0   1

1  0 0 1 | 0 = x³+1 = (x+1)(x²-x+1).

We can solve the quadratic: x=(1±√(1-4))/2=(1±i√3)/2.

The roots are therefore 1 (twice), -1, ½(1+i√3), ½(1-i√3). Two positive roots, one negative, and a pair of complex roots (conjugates).

by Top Rated User (782k points)
How did you put square root symbol? I don't see it under symbol group.

I use an iPad Air and it allows other keyboards and character sets to be added. You can get additional keyboard facilities (free apps and inexpensive apps). Not all characters can be used on this website though. Some don’t display. But many standard maths symbols are available. Look out for Unicode characters. These seem to be acceptable on this website. You can also get emojis. If you use a computer as your input device you should be able to expand your keyboard facilities using Settings. Extended physical keyboards for iPads are also available but they can be quite expensive.