The polynomial will have zeroes of
3, 4 + 2i, 4 - 2i, 1 + √7 and 1 - √7
So we have
(x - 3) (x - (4 + 2i) ) ( x - (4 - 2i) ) ( x - (1 - √7) ) ( x - (1 + √7) ) =
Breaking this down, we have
( x - (4 + 2i ) (x - ( 4 - 2i) = (x^2 -x(4 + 2i) - x (4 - 2i) + (4 + 2i) (4 - 2i) =
(x^2 -4x - 2ix - 4x + 2ix + 16 + 4) = ( x^2 - 8x + 20)
And
( x - (1 - √7)) ( x - ( 1 + √7) =
x^2 - x(1 - √7) - x ( 1 + √7) + (1 - √7) (1 + √7) =
x^2 - x + √7x - x - √7x + 1 - 7 =
(x^2 - 2x - 6)
So we have
(x - 3) ( x^2 - 8x + 20) (x^2 - 2x - 6) =
This expands to :
x^5 - 13 x^4 + 60 x^3 - 82 x^2 - 144 x + 360
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I hope this helps! :)
