Express z^4 + z^3 + z^2 + z + 1 as a product of two real quadratic factors.     this is a complex numbers question

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The given expression is,

`z^4+z^3+z^2+z+1`

If we want to write this as a product of two quadratic expressions, the following equation satisfies our requirement.

`z^4+z^3+z^2+z+1 = (z^2+az+1)(z^2+bz+1)`

a and b are constants which we have to determine later.

Expanding the RHS,

`z^4+z^3+z^2+z+1 = z^4+(a+b)z^3+(1+ab+1)z^2+(a+b)z+1`

Now we can compare the coefficients on either side to find values for a and b.

Through coefficient of `z^3` terms, we get,

`a+b = 1`      -----------> Eq:1

Through coefficient of `z^2` terms, we get,

`1+ab+1 = 1`

`ab = -1` ------------------> Eq:2

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