Question

How do I solve this polynomial?

the equation is: `x^4+4x^2+2=0` when asking for how many solutions are distinct and all the `x` belonging to `CC` (complex). Also expressing all solutions in polar form and for one solution of `x` finding the exact value of `x^55` and the modulus

Answer 1

Given expression

`x^4+4x^2+2=0`

We see that the above equation can not be factorized using the split the middle term.

However we note that the equation can be reduced to a quadratic. using substitution

`w = x^2`

Thus transforming it as

`w^2+4w+2 = 0`

Roots of this quadratic are

`w=(-4+-sqrt(4^2-4xx1xx2))/(2xx1)`
`w=-2+-sqrt(2)`

are two real solutions but both are negative. Now finding values of `x` we get

`x=+-sqrt(-2+-sqrt(2))`

all four values of `x` belong to `CC`

.-.-.-.-.-.-.-.-.-

Approximately `w= -0.5858 or -3.4142`
`:.` Four solutions can be written in the complex form (`a+bi`) as

`x =+-sqrt(-3.414) = 0.0 - 1.84776 i or 0.0 + 1.84776 i`

`x =+-sqrt(\-0.586) = 0.0 - 0.76537 i or 0.0 + 0.76537 i`