There is no real consensus in the literature about what exactly constitutes the fundamental theorem of algebra; it is stated differently in different texts. The two-part theorem stated in this lesson encapsulates the main ideas of the theorem and its corollaries while remaining accessible to students. The first part of what is stated here as the fundamental theorem of algebra is the one that students are not mathematically equipped to prove or justify at this level; this part states that every polynomial equation has at least one solution in the complex numbers and will need to be accepted without proof.
The consequence of this first part is what is really interesting—that every polynomial expression factors into the same number of linear factors as its degree. Justification for this second part of the fundamental theorem of algebra is accessible for students as long as they can accept the first part without needing proof. Since every polynomial of degree 𝑛 ≥ 1 will factor into 𝑛 linear factors, then any polynomial function of degree 𝑛 will have 𝑛 zeros (including repeated zeros).
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