First of all: a polynomial of degree N has exactly N roots (in the complex plane) [ref]

Second: since for every root there is also its complex-conjugate [ref], we can only have some combinations...

Degree | Roots | Possible Combinations |

1 | 1 | Real Root |

2 | 2 | Real Roots, or 2 Complex Roots |

3 | 3 | Real Roots, or 1 Real and 2 Complex Roots |

4 | 4 | Real Roots, or 2 Real and 2 Complex Roots, or 4 Complex Roots |

etc etc! |

Eventually, complex numbers are defined like this.

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