What are the roots of x^4-12x^3+31x^2+12x-32  = 0

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justaguide eNotes educator| Certified Educator

The roots of the equation x^4-12x^3+31x^2+12x-32  = 0 have to be determined.

x^4-12x^3+31x^2+12x-32  = 0

=> x^4 - x^3 - 11x^3 + 11x^2 + 20x^2 - 20x + 32x - 32 = 0

=> x^3(x - 1) - 11x^2(x - 1) + 20x(x - 1) + 32(x - 1) = 0

=> (x^3 - 11x^2 + 20x + 32)(x - 1) = 0

=> (x^3 + x^2 - 12x^2 - 12x + 32x + 32)(x - 1) = 0

=> (x^2(x + 1) - 12x( x + 1) + 32(x + 1))(x - 1) = 0

=> (x^2 - 12x + 32)(x - 1)(x + 1) = 0

=> (x^2 - 8x - 4x + 32)(x - 1)(x + 1) = 0

=> (x(x - 8) - 4(x - 8))(x - 1)(x + 1) = 0

=> (x - 4)(x - 8)(x - 1)(x + 1) = 0

=> x = 4, x = 8, x = 1 and x = -1

The roots of the equation x^4-12x^3+31x^2+12x-32  = 0 are {-1, 1, 4, 8}