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This essential trains for: SAT-II, AMC-10, AMC-12, AIME, Math Kangaroo 9-10.

The coefficients and the roots of polynomial equations with complex coefficients are related. The equations describing how they relate are called Viète's relations.

For a polynomial equation of degree n:

equation

with roots:

equation

the canonical expression and the factored expression must be identical:

equation

Using the definition of equal polynomials:

equation

equation

equation

equation

equation

Note that the system of equations with unknowns x1, x2,..., xn formed by the relations is equivalent to the polynomial equation. The complexity of solving the system is the same as the complexity of solving the equation. Therefore, there is no advantage in solving the system over solving the equation itself.

However, the relations may be very useful in answering auxiliary questions about the equation.