+128475 x^4 - 3x^3 + 10x^2 - 24 + 16 = 0
Note that if we can add the coefficients and the constant and we get 0 , then 1 is a root
1 + 10 + 16 - 3 - 24 =
27 - 27 = 0
So 1 is a root
Using some synthetic division to find the reduced polynomial, we have
1 [ 1 - 3 10 -24 16 ]
1 -2 8 -16
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1 -2 8 -16 0
So....the reduced polynomial is
x^2 - 2x^2 + 8x - 16 = 0 this factors as
x (x -2) + 8 ( x -2) = 0
(x + 8) ( x -2) = 0
Setting each factor to 0 and solving for x gives us
x = -8 and x = 2
So the roots are
x = -8 x = 1 and x = 2
I'm not sure but after the synthetic division, wouldn't the equation be x^3 instead of x^2? Because if you look at the original equation, it starts with x^4, not x^3. So after recollecting the numbers, the equation would be
x^3 - 2x^2 + 8x -16 = 0
From here, it would factor out as,
x^2(x-2) +8(x-2) = 0
(x-2) (x^2 +8) = 0
With the zero product property,
x - 2 = 0
x = 2
x^2 +8 = 0
x^2 = -8
After sqaure rooting each side,
x = 2 sqrt 2i, x = -2 sqrt 2i
So, in total, all the roots would be x = 1, 2, 2 sqrt 2i and -2 sqrt 2i
