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All courses Math II · N-CN.9 59 of 73
Know the Fundamental Theorem of Algebra and verify it for quadratic polynomials.

Determine how many complex roots a quadratic polynomial has, counting multiplicity

Problem
How many complex roots does x²-5x+6 have, counting multiplicity?
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Hint

Notice the highest power of x in x²-5x+6. Use the degree with the Fundamental Theorem of Algebra.

The Fundamental Theorem of Algebra counts roots with multiplicity, and real roots are included among complex roots.

Solution walkthrough

01

Identify the polynomial degree

\[x^2-5x+6~\text{has}~\text{degree}~2\]

The highest exponent with a nonzero coefficient is 2.

02

Apply the Fundamental Theorem of Algebra

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

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

03

Use n=2

\[\text{degree}~2~->~2~\text{complex}~\text{roots}~\text{counting}~\text{multiplicity}\]

Real roots are included in the complex-number count.

04

Factor to check the count

\[x^2-5x+6=(x-2)(x-3)\]

The explicit roots 2 and 3 are two distinct real numbers and therefore also two complex roots.

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Another way

  1. Factor first to see the two roots directly, then note that multiplicity total is 1+1=2.

!

Common mistake

Do not exclude real roots from the complex count. Every real number is also a complex number with imaginary part zero.