Question

How do you determine the number of complex roots of a polynomial of degree$n$?

Answer 1

In this question we need to find the number of complex roots of a polynomial of degree$n$. To solve this we need to know that real numbers are also complex numbers because real numbers are subsets of complex numbers. Also knowledge that quadratic equation which is polynomial of degree 2 and has 2 roots, cubic equation which is a polynomial of degree 3 has 3 roots and so on.

Complete step by step solution:
Let us try to find the solution to the above question. In this question we need not to solve, answer to this question is depend on a famous theorem named Fundamental theorem of algebra which states that “single variable, non-constant polynomial which has complex (real also because every real number is complex number) coefficients has at least one complex root.”

The theorem can be stated as “every non-zero, single variable, degree $n$polynomial with complex (real also because every real number is complex number) coefficients, counted with multiplicity, exactly complex roots.”

Hence by using the fundamental theorem of algebra a polynomial of degree $n$has exactly $n$complex roots counting multiplicity.

Note: This is a theoretical question based on the number of polynomials of degree$n$. Fundamental theorem of algebra is a very important place in higher mathematics. Proof of this theorem requires knowledge of higher mathematics. So we prove this theorem from our current knowledge of mathematics. Just knowing the result is more than sufficient.