# Roots and polynomials

If z is an n-th term of 1, show taht 1+z+z^2+z^3+....+z^n-1=0.

Solve the equation with n=5 and hence factorize 1+z+z^2+z^3+z^4 into linear factors with complex coefficient and then into quadratic factors with real coefficients.

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

<br> =z^4+(a+b)z^3+(ab+2)z^2+(a+b)z+1

<br> So we have a+b=1, ab+2=1 or ab=-1

<br> Thus a and b satisfies the quadratic equation x^2-x-1=0

<br> By quadratic formula, we have a=(1+sqrt(5))/2, b=(1-sqrt(5))/2

<br> Therefore,

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