# 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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#### Solution Preview

Proof: Since z is an n-th root of 1, z!=1 (!= means not equal to), we have 1-z^n=0. Note 1-z^n=(1-z)(1+z+z^2+...+z^(n-1)) and 1-z!=1, thus we have 1+z+z^2+...+z^(n-1)=0.

<br>Now suppose n=5. we want to factorize 1+z+z^2+z^3+z^4.

<br>(1)quadratic factors with real coefficients

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