What's the minimum order of the polynomial?

Question

A polynomial has one unique real root, a set of repeating roots, and 2 complex roots. What's the minimum order of the polynomial? The order of a polynomial is the degree of the exponent of the leading term.

Answer 1

"Repeating roots" means we have a root of
multiplicity greater than 1.  This requires a
factor of (x-z))ⁿ where n>1
To be of minimum degree n=2
 
x(x-z)ⁿ(x-(a+bi))(x-(a-bi))
 
x(x-z)²(x-a-bi)(x-a+bi)
x(x-z)(x-z)(x-a-bi)(x-a+bi)
 
Multiplied out this will be a polynomial of 5th degree

Answer 2

2 complex roots gives a degree of 2 (quadratic).  One unique root (linear) added to that makes it a degree higher.  You are now at degree 3.  Then a set of repeating roots could be any degree (x - c)^r.  In this case, r is any value greater than or equal to 2.  So the degree in this case is at least 2.

 

You can have a minimum degree of 5.  No less than that.

Answer 3

What you've called order here is usually called DEGREE of the polynomial.

 

"A set of repeating roots" is vague--does it say that (x-r)ⁿ is a factor, and that the multiplicity of root r is n?

As you see, since n is unknown, the degree cannot be determined.  

 

n has to be ate least two, for there to be repeating roots; then the minimum degree must be 5, because of

a complex conjugate pair of roots, one unique real root, and at least n=2 for the minimal # of repeated roots.