Find the equation of the 4th degree polynomial with real coe

Find the equation of the 4th degree polynomial with real coefficients with roots 3, 0 and 1 - i. Make sure to expand your answer (do not leave in factored form).

Solution

We know that imaginary roots always occurs in pairs. So if one rp=oot is 1-i then there must be the second root 1+i (conjugate of 1-i)

Hence we have roots

3, 0, 1-i and 1+i

therefore the polynomial is

(x-0) (x-3) (x-(1-i)) (x-(1+i))

= (x^2 -3x) (x-1+i) (x-1-i)

= (x^2 -3x) ( (x-1)^2 - i^2))

= (x^2 -3x) (x^2 -2x +1 - (-1))

= (x^2 -3x) (x^2 -2x +2)

= x^4 -5x^3 +8x^2 -6x