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. 3- i, i, and 2

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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.  3- i, i, and 2

usaho4w · Accepted Answer

Step 1  Given:  The zeros of the polynomial 3-i, i and 2  Step 2  The complex roots occur in pair, so conjugate of 3-i and i also be the zeros of the same polynomial  So, the polynomial has zeros: 3-i, 3+i, i , -i and 2  If a1,a2,a3,a4,a5 are the zeros of the polynomial then the polynomial in factored form is written as:  f(x)=a(x−a1)(x−a2)(x−a3)(x−a4)(x−a5)  Where a is the leading coefficient since the given polynomial is monic so the leading coefficient is 1.  ⇒f(x)=1(x−(3−i))(x−(3+i))(x−(i))(x−(−i))(x−2)  ⇒f(x)=1(x−3+i)(x−3−i)(x−i)(x+i)(x−2)  Simplifying we get (x−3+i)(x−3−i)=x2−6x+10  (x−i)(x+i)=x2+1  ⇒f(x)=(x2−6x+10)(x2+1)(x−2)  ⇒f(x)=x5−8x4+23x3−28x2+22x−20  Therefore, the required monic polynomial is f(x)=x5−8x4+23x3−28x2+22x−20

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

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