GOZUNTA
Try it now
All courses Algebra II · 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: \(\text{degree}~4\) polynomial?
Your answer
Show answer choicesHide answer choices Work through the mathematics first, then compare your reasoning.Select the answer that matches your work.
With a free account

Let your results add up.

Each saved attempt adds useful evidence about what you have practiced and what deserves another pass.

Create a free account
With paid access

Follow a skill back to its prerequisites.

All-course access lets you move between Math Foundations and Math I–III when an earlier idea needs attention.

Compare plans

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 polynomial degree

\[\text{degree}=4\]

The prompt states that the polynomial is degree 4.

02

Apply the Fundamental Theorem of Algebra

\[\text{degree}~n~->~n~\text{complex}~\text{roots}~\text{counted}~\text{with}~\text{multiplicity}\]

Every nonconstant degree-n polynomial over the complex numbers has exactly n roots when repeated roots are counted.

03

Substitute the degree

\[n=4~->~\text{total}~\text{complex}-\text{root}~\text{count}=4\]

The count includes real roots because real numbers are also complex numbers.

04

State the precise answer

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

Distinct-root count could be smaller if roots repeat, but the multiplicity count is always four.

+

Another way

  1. Factor conceptually into four complex linear factors, allowing repeated factors; each factor supplies one root occurrence.

!

Common mistake

Do not claim four distinct roots. The theorem guarantees four roots counted with multiplicity, so repeated roots still contribute to the total.

Solution walkthrough video