Find a monic polynomial f(x) of least degree over C

Daniell Phillips

Daniell Phillips

Answered question

2021-12-19

Find a monic polynomial f(x) of least degree over C that has the given numbers as zeros, and a monic polynomial g(x) of least degree with real coefficients that has the given numbers as zeros.
i, 2-i

Answer & Explanation

temnimam2

temnimam2

Beginner2021-12-20Added 36 answers

Given
we have to find a monic polynomial whose zeros are x=i and x=2-i
solution
f(x)=anxn+an1xn1++a1x+a0 be a monic polynomial then an=1
polynomial has a complex root x=a+ib then its conjugate also a root x=a-ib
given,
x=i implies x=-i also a root
(x+i)(xi)=x2i2=x2+1
x=2-i implies x=2+i also a root
(x-(2-i))(x-(2+i))=(x-2+i)(x-2-i)
=x22xix2x+4+2i+ix2ii2
=x24x+5
now,
f(x)=(x2+1)(x24x+5)
=x44x3+5x2+x24x+5
f(x)=x44x3+6x24x+5 is the required monic polynomial
Cleveland Walters

Cleveland Walters

Beginner2021-12-21Added 40 answers

Step 1
It is known that complex zeros of a polynomial with real coefficients occurs in pairs.
Given that i and 2-i are complex zeros of polynomial.
Therefore, -i and 2+i are also zeros of same polynomial.
Step 2
A least degree polynomial over R can be obtained as follows.
f(x)=(x-i)(x+i)(x-(2-i))(x-(2+i)))
=(x2i2)((x2)2i2)
=(x2+1)((x2)2+1)
Thus, a required least degree polynomial over R is f(x)=(x2+1)((x2)2+1).
A required least degree polynomial over C is g(x)=(x-i)(x-2+i).

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?