A polynomial a_nzn+a_n-1zn-1+. s+a_1z+a_0, where a_n 0, has exactly n roots in C, counted with multi

Question

A polynomial a_nzn+a_n-1zn-1+. s+a_1z+a_0, where a_n 0, has exactly n roots in C, counted with multiplicity. If a polynomial has real coefficients and z is a non-real root, then z* is also a root with the same multiplicity. Therefore non-real roots of a real-coefficient polynomial occur in conjugate pairs. **Degree** | **Real roots** | **Non-real complex roots** | **Total** 2 (quadratic) | 2 | 0 | 2 | 0 | 2 | 3 (cubic) | 3 | 0 | 3 | 1 | 2 | 4 (quartic) | 4 | 0 | 4 | 2 | 2 | | 0 | 4 | 5 (quintic) | 5 | 0 | 5 | 3 | 2 | | 1 | 4 | The counts include multiplicity. A repeated real root is counted each time it occurs. A number alpha is a root of P(z)=0 precisely when substituting it gives P(alpha)=0.

Accepted Answer

$\deg P=n\Rightarrow n$ roots counted with multiplicity; for real coefficients, non-real roots occur in conjugate pairs $z,z^*$.

Polynomial Roots and Conjugate Pairs

Polynomial Roots and Conjugate Pairs

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Degree	Real roots	Non-real complex roots	Total
2 (quadratic)	2	0	2
2 (quadratic)	0	2	2
3 (cubic)	3	0	3
3 (cubic)	1	2	3
4 (quartic)	4	0	4
	2	2
	0	4
5 (quintic)	5	0	5
	3	2
	1	4