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

Daniell Phillips 2021-12-19 Answered
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
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temnimam2
Answered 2021-12-20 Author has 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

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Cleveland Walters
Answered 2021-12-21 Author has 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).

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