what is the least possible degree of the function?

Question

A polynomial function with real coefficients has zeros at -2, -1, √2, and -3i. What is the least possible degree of the function?

 

 

A polynomial function with rational coefficients has zeros at -2, -1, √2, and -3i. What is the least possible degree of the function?

Answer 1

The first polynomial must be at least of degree 5. Coefficients could be irrational, but at least they would be real.  For the second case, the minimum degree would be 6.  In that case, the congugate of both the irrational root and imaginary root must be included.

Answer 2

The factors of the polynomial are:

 

(x + 2)(x + 1)(x - √2)(x + √2)(x + 3i)(x - 3i)              irrational and complex roots come in conjugate pairs

 

When multiplied out the polynomial shall have a degree of 6.