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

David Young

David Young

Answered question

2021-12-11

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.
2, 1-i

Answer & Explanation

encolatgehu

encolatgehu

Beginner2021-12-12Added 27 answers

Step 1
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.
2, 1 - i.
The zeros of the polynomial are 2 and 1-i.
The complex roots occurs in pairs so conjugate of 1-i, that is, 1+i is also a zero of the polynomial.
Step 2
So, the polynomial f(x) has zeros 2, 1-i, 1+i.
Then f(x) can be written as
f(x)=a(xa1)(xa2)(xa3)
where a is the leading coefficient, given that the polynomial is monic, we have the leading coefficient to be 1.
Therefore, we have,
f(x)=1(x-2)(x-(1-i))(x-(1+i))
=(x-2)((x-1)+i)((x-1)-i)
=(x2)((x1)2i2)
=(x2)(x22x+1+1)
=x(x22x+2)2(x22x+2)
=x32x2+2x2x2+4x4
=x34x2+6x4

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