Determine the product of all real roots of the equation (x^4)+((x-4)^4)=32

Question

This is sooooo hard

Answer 1

(x-4)^4=

(x^2 - 8x + 16)^2 =

(x^2 - 8x + 16)(x^2 - 8x + 16) =

x^4 - 8x^3 + 16x^2
- 8x^3 + 64x^2 - 128x
16x^2 - 128x + 256

---------------------------------

x^4 - 16x^3 + 96x^2 - 256x + 256




2x^4 - 16x^3 + 96x^2 - 256x + 256 = 32

2x^4 - 16x^3 + 96x^2 - 256x + 224 = 0

x^4 - 8x^3 + 48x^2 - 128x + 112 = 0


By rational root theorem, the possible rational
roots are +or- {1,2,4,7,8,14,16,28,56,112}

X=2 is a solution.

Synthetic division 2 | 1 -8 48 -128 112
2 -12 72 -112
----------------------------
1 -6 36 -56 0

x^3 - 6x^2 + 36x - 56 = 0

Again, by rational root theorem, the possible rational
roots are +or- { 1, 2, 4, 7, 8, 14, 28, 56}

Synthethic division:

2 | 1 -6 36 -56
2 -8 56
---------------------
1 -4 28 0

x^2 - 4x + 28 = 0

x = [4 +or- sqrt( -96) ]/2

x = [4 +or- 4*Sqrt(6)]/2

= 2 +or- 2*Sqrt(6)


The product of the roots is 2 * 2 * ( 2 + 2*i*Sqrt(6)) * (2 - 2*i*sqrt(6))

4 * ( 4 + 24)

112