Find the polynomial with rational coefficients whose two zeros are #3-i# and #sqrt7#?

1 Answer
Dec 7, 2017

Polynomial is #x^4+3x^2-70#

Explanation:

When a polynomial functional has rational coefficients,

  1. its complex zeroes are in pairs and complex conjugates of type #a+-bi#
  2. its irrational roots are in pairs and conjugate real numbers of the type #+-sqrtz#

and if #alpha,beta,gamma,delta# are all the roots, none of them repeating, then its lowest degree should be #4# and rational polynomial would be #(x-alpha)(x-beta)(x-gamma)(x-delta)#. So if there are seven roots degree, lowest degree would be #7#.

Here we have #3-i# and #sqrt7# as two roots and as it has rational roots, other roots must be #3+i# and #-sqrt7# and polynomial would be

#(x-(3-i))(x-(3+i))(x-sqrt7)(x-(-sqrt7)#

= #(x-3+i)(x-3-i)(x-sqrt7)(x+sqrt7)#

= #(x^2+9-i^2+ix-ix+3i-3i)(x^2-7)#

= #(x^2+10)(x^2-7)# - as #-i^2=1#

= #x^4+3x^2-70#