Zeros and functions

Question

Find a polynomial of lowest degree with rational coefficents that has the given numbers as some of its zeros.

The zeros are the square root of 3, and 5i

Thanks!

Answer 1

Since the polynomial has rational coefficients, complex roots must occur in conjugate pairs.  

 

So, since 5i is a root, so is -5i.

 

Also, we can't have square roots of non-perfect squares as coefficients.

 

So, let's use -√3 as a root (because √3(-√3) is rational) 

 

For each root, c, x-c is a factor.

 

So, a polynomial of smallest degree is (x-√3)(x+√3)(x-5i)(x+5i)

 

                                                  = (x²-3)(x²+25)

 

                                                  = x⁴+22x²-75