Homework Help

What is the simplest real polynomial with roots 1 + i and 2 - i

user profile pic

x32 | Student | (Level 1) eNoter

Posted August 10, 2013 at 3:37 PM via web

dislike 1 like

What is the simplest real polynomial with roots 1 + i and 2 - i

1 Answer | Add Yours

user profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted August 10, 2013 at 3:41 PM (Answer #1)

dislike 1 like

The polynomial has roots 1 + i and 2 - i. As it has real coefficients the roots are present in pairs of complex conjugate numbers. The simplest polynomial with these roots would also have 1 - i and 2 + i as its roots. This gives the simplest polynomial as:

(x - 1 - i)(x - 1 + i)(x - 2 + i)(x - 2 - i)

= ((x - 1)^2 + 1)((x - 2)^2 + 1)

= x^4 - 6x^3 + 15x^2 - 18x + 10

The required polynomial is x^4 - 6x^3 + 15x^2 - 18x + 10

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes