polynomial function of least degree that has rational coefficients, leading coefficient of 1, and the given. Write the polynomial in standard form. 3i, 2−i

Question

I need help, I don't understand

Answer 1

The idea here is probably that 3i and 2 -i are zeros of the desired polynomial.   Since the coefficients are rational, the complex zeros come in complex conjugate pairs.  This means that  -3i and  2+ i are also zeros.  A start on the polynomial is to write

 

   P(x) = (x - 3i) (x + 3i) (x -(2-i) )( x - (2 +i))

 

   Multiplying this all out results in

 

  P(x) =  x⁴  + 14 x²  + 45

 

Since the coefficient of the leading term is 1,   this is the final answer.