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All courses Math III · N-CN.9 46 of 55
Know the Fundamental Theorem of Algebra and connect it to polynomial roots.

Determine the total number of complex roots of a polynomial from its degree

Problem
How many complex roots does this polynomial have in total: degree 4 polynomial?
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Hint

Use the degree of the polynomial to determine the total number of complex roots.

By the Fundamental Theorem of Algebra, a polynomial of degree \(n\) has exactly \(n\) complex roots when counted with multiplicity.

Solution walkthrough

01

Identify the degree

\[4\]

The question says the polynomial has degree 4.

02

Apply the Fundamental Theorem of Algebra

\[\text{degree}~n~\text{polynomial}~\to~n~\text{complex}~\text{roots}\]

A polynomial has as many complex roots as its degree when multiplicity is counted.

03

Substitute the degree

\[n~=~4\]

So this polynomial has 4 complex roots in total.

04

State the total root count

\[4~\text{complex}~\text{roots}~\text{counted}~\text{with}~\text{multiplicity}\]

This includes real and nonreal roots, and repeated roots count more than once.

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Another way

  1. You can remember this as: the degree tells the total root count over the complex numbers.

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Common mistake

A common mistake is to count only real roots, but the question asks for total complex roots, including nonreal ones.