# Explain how to solve a polynomial equation finding all zeros including complex zeros.

sciencesolve | Teacher | (Level 3) Educator Emeritus

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The problem does not specify the polynomial, hence you need to remember several things when you try to find all the zeroes of a polynomial.

If the equation is a quadratic, you may use the quadratic formula to find the roots such that:

x_(1,2) = (-b+-sqrt(b^2-4ac))/(2a)

a,b,c express the coefficients of quadratic equation

The roots x_1 and x_2 may be real or complex as expression b^2-4ac is positive or negative.

If the polynomial degree is larger than two, then there are altternative methods used for finding the roots. You need to remember that the number of roots needs to be equal to the highest order power of variable.

You may group the terms and use the factorization to find the ou may use roots or if the factorization method fails, you may use rational roots test.

To perform the rational roots test, you should collect all factors of leading coefficient and constant term and you need to form fractions such that the numerator will be one factor of constant term and the denominator will be one factor of leading coefficient.

You need to substitute these farctions for x in the polynomial to verify if they cancel the polynomial.

If one fraction cancels the polynomial, it becomes a zero for polynomial.

The process of finding the roots should continue using this guessing method or you may use the reminder theorem to find the quotient. You may find the quotient roots using one of the methods described above. The method you will select depends on the polynomial degree.

Hence, you need to remember that the polynomial degree and the terms involved in polynomial should help you to select  what is the appropriate method for finding the roots.