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@johnstanley5692 Ай бұрын
multiply by x^5 and let y=2*x to obtain p(y) = y^5-y^4+y^3-y^2-y+y-1 =0, now use (y^6+1)=(y+1)*p(y)=0; Hence need solutions of y^6=-1 and hence y = exp(i*(2*n+1)*pi), n=0:5, now use x=(y/2) to obtain the 6 values of xi --> {+/-0.5*i, 0.433+/-0.25*i, -0.433+/-0.25*i}
@SidneiMV Ай бұрын
Geometric Progression common ratio = -1/(2x) 32 - 16/x + 8/x² - 4/x³ + 2/x⁴ - 1/x⁵ = 0 32 + 32(-1/(2x)) + 32(-1/(2x))² + 32(-1/(2x))³ + 32(-1/(2x))⁴ + 32(-1/(2x))⁵ = 0 1 + (-1/(2x)) + (-1/(2x))² + (-1/(2x))³ + (-1/(2x))⁴ + (-1/(2x))⁵ = 0 -1/(2x) = u => x = -1/(2u) 1 + u + u² + u³ + u⁴ + u⁵ = 0 1 + u + u² + u³ + u⁴ + u⁵ = (u⁶ - 1)/(u - 1) (u⁶ - 1)/(u - 1) = 0 (u ≠ 1) *[ I ]* u⁶ - 1 = 0 => u⁶ = 1 (-1/(2x))⁶ = 1 => 1/(64x⁶) = 1 64x⁶ = 1 => x⁶ = (1/2)⁶ => *x = 1/2 (for real values)* (x = -1/2 not valid because of "I")
@ronbannon Ай бұрын
If you solve this using the geometric pattern, you will get the exact five solutions as presented, but an extraneous root x = -1/2 will also be seen. The geometric pattern reduces to a sixth-degree polynomial, x^6 = 1/64. Using de Moivre's formula, you will find six solutions.
@SidneiMV Ай бұрын
@@ronbannon Your comment is interesting. So I edited my solution (comment) explaining why x = -1/2 is not a solution.
@johnstanley5692 Ай бұрын
whoops (y^6-1)= (y+1)*p(y)=0, so need solutions of y^6=+1 => yi =exp(i*2*n*pi), n=0:5, again use x=y/2. => xi= { +/-0.5, 0.25+/-0.433*i, -0.25+/-0.433i}
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