Let fx x4 3x3 27x2 387x 2044 Owen that 3 8i is a root

Let f(x) = x^4 - 3x^3 + 27x^2 + 387x - 2044 Owen that 3 + 8i is a root of f(x), list all of the roots including the complex roots.

Solution

f(x)=x⁴-3x³+27x²+387x-2044

given 3+8i is root

since all the coefficients are real conjugate 3-8i is also the root

(x-(3+8i))(x-(3-8i))

=((x-3)-8i)((x-3)+8i)

=(x-3)²-(8i)²

=(x²-6x+9)-(-64)

=x²-6x+73

for other factors divide f(x) by x²-6x+73

x²-6x+73] x⁴-3x³+27x²+387x-2044 [x²+3x-28
^{...................}x⁴-6x³+73x^{2
.............................................................................
.......................}3x³-46x²+387x
^{.......................}3x³-18x²+216x
^{............................................................................
............................}-28x²+171x-2044

^{............................}-28x²+171x-2044

^{..............................................................
...........................................0}

factorise x²+3x-28

x²+7x-4x-28

(x+7)(x-4)

x⁴-3x³+27x²+387x-2044 =(x-(3-8i))(x-(3+8i))(x+7)(x-4)

for roots f(x)=0

(x-(3-8i))(x-(3+8i))(x+7)(x-4)=0

x =3-8i,x =3+8i ,x =-7,x =4