does the degree of a polynomial match the number of distinct roots?

Question

please help me anwser this question thank you very much

Answer 1

The number of distinct roots cannot exceed the degree of the polynomial.  It is always less than or equal to the degree.  As the tutors indicated, some roots may be identical, so that reduces the number of distinct roots in some cases.

Answer 2

NO

 

As a counter-example,

 

x^2 -2X +1  has two identical roots x=-1 because it factors like this: (x-1)(x-1)

 

Yet it has degree 2, but only 1 distinct root.

 

Surely you can think of many more.

Answer 3

Not necessarily.  For example, the polynomial x²+2x+1 has degree 2 but has the double root -1. So, the number of distinct roots is smaller than the degree of the polynomial.