Polynomial question

Question

If (X-1/X)²+(X+1/X)²=16, find the value of X⁶+1/X⁶

Answer 1

If you expand the binomials you get 2[x²+(1/x²)]=16

If x is not zero, this will give you a quadratic in x² which you can solve with the quadratic formula and then take the square root of the result.

Answer 2

Squaring out the left hand side of the given equation gives

x² - 2 + 1/x² + x² + 2 + 1/x² = 16 which simplifies to

2x² + 2/x² = 16 and dividing by 2 gives

x² + 1/x² = 8.

Squaring both sides of this result gives

(x² + 1/x²)² = 8² or

x⁴ + 1 + 1/x⁴ = 64 or, subtracting 2 from both sides, (see why later),

x⁴ - 1 + 1/x⁴ = 62.

Factoring x⁶ + 1/x⁶ using the sum of cubes formula gives

x⁶ + 1/x⁶ = (x² + 1/x²)(x⁴ - 1 + 1/x⁴) and substituting the above values gives the result

x⁶ + 1/x⁶ = (x² + 1/x²)(x⁴ - 1 + 1/x⁴) = (8)(62) = 496. Hope this helped!