Express z^4 + z^3 + z^2 + z + 1 as a product of two real quadratic factors. this is a complex numbers question
The given expression is,
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
(The entire section contains 290 words.)
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