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Q01 |
- Diagram to Solve x^5 + 1 = 0
- Q02 | - DeMovire's theory
- Q03 | - Solve x^5 + 1 = 0 by DeMovire's theory
- Q04 | - Solve x^5 + 1 = 0 by construction
- Q05 | - Solve x^4 - x^3 + x^2 - x + 1 = 0
- Q06 | - Reference
- Q02 | - DeMovire's theory
Q01. Diagram :
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Q02. DeMovire's thoery
Rule 1
- (cos(A) + i*sin(A))^n = cos(n*A) + i*sin(n*A)
- (cos(A) + i*sin(A))^(1/n) = cos((2*k*pi + A)/n) + i*sin((2*k*pi + A)/n)
- Where k = 0, 1, 2, .... (n - 1)
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Q03. Solve x^5 + 1 = 0 by DeMovire's theory
Change x^5 = -1 to polar form
- x^5 = cos(180) + i*sin(180)
- x = (cos(180) + i*sin(180))^(1/5)
- x = cos((2*k*pi + 180)/5) + i*sin((2*k*pi + 180)/5)
- k = 0, x0 = cos(036) + i*sin(036)
- k = 1, x1 = cos(108) + i*sin(108)
- k = 2, x2 = cos(180) + i*sin(180) = -1
- k = 3, x3 = cos(252) + i*sin(252)
- k = 4, x4 = cos(324) + i*sin(324)
- x0 and x4 are conjugate
- x0 + x4 = cos(36) + i*sin(36) + cos(324) + i*sin(324)
- = cos(36) + i*sin(36) + cos(360 - 36) + i*sin(360 - 36)
- = cos(36) + i*sin(36) + cos(36) - i*sin(36)
- = 2*cos(36)
- = Real number
- x0*x4 = (cos(36) + i*sin(36))*(cos(324) + i*sin(324))
- = (cos(36) + i*sin(36))*(cos(36) - i*sin(36))
- = cos(36)^2 - (i^2)*sin(36)^2
- = cos(36)^2 + sin(36)^2
- = 1
- = Real number
- x1 and x3 are conjugate
- x1 + x3 = cos(108) + i*sin(108) + cos(252) + i*sin(252)
- = cos(180 - 72) + i*sin(180 - 72) + cos(180 + 72) + i*sin(180 + 36)
- = -cos(72) + i*sin(72) - cos(72) - i*sin(72)
- = -2*cos(72)
- = Real number
- x0*x4 = (cos(108) + i*sin(108))*(cos(252) + i*sin(252))
- = (-cos(72) + i*sin(72))*(-cos(72) - i*sin(72))
- = cos(72)^2 - (i^2)*sin(72)^2
- = cos(72)^2 + sin(72)^2
- = 1
- = Real number
- sin(180 - A) = +sin(A)
- sin(180 + A) = -sin(A)
- sin(360 - A) = -sin(A)
- cos(180 - A) = -cos(A)
- cos(180 + A) = -cos(A)
- cos(360 - A) = +cos(A)
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Q04. Solve x^5 + 1 = 0 by construction
Construction
- Draw a large unit circle (Radius = 1 unit)
- Draw five points P, Q, R, S, T on circle
- Draw angle A0 = angle POX = 036
- Draw angle A1 = angle QOX = 108
- Draw angle A2 = angle ROX = 180
- Draw angle A3 = angle SOX = 252
- Draw angle A4 = angle TOX = 324
- Find root r0
- Let coordinates P be (x0, y0)
- Measure x0 and y0 then r0 = x0 + i*y0
- Find root r1
- Let coordinates Q be (x1, y1)
- Measure x0 and y0 then r1 = x1 + i*y1
- r2 = -1
- r3 is conjugate of r1
- r4 is conjugate of r0
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Q05. Solve x^4 - x^3 + x^2 - x + 1 = 0
Use solution of x^5 + 1 = 0
- Since x^5 + 1 = (x + 1)*(x^4 - x^3 + x^2 - x + 1)
- The roots are r0, r1, r3, r4
- r0 = cos(036) + i*sin(036)
- r1 = cos(108) + i*sin(108)
- r3 = cos(252) + i*sin(252)
- r4 = cos(324) + i*sin(324)
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Q06. Reference
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