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Mathematics Dictionary
Dr. K. G. Shih
Power 6 Equation
Questions

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  • Q01 | - Method 1 : Solve x^6 + x^5 - 6*x^4 - 7*x^3 - 6*x^2 + x + 1 = 0
  • Q02 | - Method 2 : Solve x^6 + x^5 - 6*x^4 - 7*x^3 - 6*x^2 + x + 1 = 0
  • Q03 | - Method 3 : Solve x^6 + x^5 - 6*x^4 - 7*x^3 - 6*x^2 + x + 1 = 0
  • Q04 | - Method 1 : Solve x^6 + x^5 - 6*x^4 - 7*x^3 - 6*x^2 + x + 1 = 0
  • Q05 | - Method 1 : Solve x^6 + x^5 - 6*x^4 - 7*x^3 - 6*x^2 + x + 1 = 0
  • Q06 | - Comments : Solve x^6 + x^5 - 6*x^4 - 7*x^3 - 6*x^2 + x + 1 = 0
  • Q07 | - Solve x^7 + 1 = 0
  • Q08 | - Solve x^7 - 1 = 0
  • Q09 | -
  • Q10 | -
Answers


Q01. Method 1 : Solve x^6 + x^5 - 6*x^4 - 7*x^3 - 6*x^2 + x + 1 = 0

Graphic solution
  • Study Program | Graphic Solutions of Polynomial Functions.
  • Use program 01 11, we get the following real roots by estimation
    • 0.382, -0.382, 2,618 and -2.618 (Approx graphic solutions of program 01 11)

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Q02. Method 2 : Solve x^6 + x^5 - 6*x^4 - 7*x^3 - 6*x^2 + x + 1 = 0

Synthetic Division
  • 1 +1 -6 -7 -6 +1 +1 | -1
  • . -1 -0 +6 +1 +5 -6
  • --------------------------
  • 1 +0 -6 -1 -5 +6 -5
  • The remaider is -5, hence x = -1 is not a solution
  • 1 +1 -6 -07 -06 +01 +01 | +1
  • . +1 +2 -04 -11 -17 -16
  • --------------------------
  • 1 +2 -4 -11 -17 -16 -15
  • The remaider is -15 hence x = 1 is not a solution
  • But other real solutions are not integers and are not easy to find.

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Q03. Method 2 Solve x^6 + x^5 - 6*x^4 - 7*x^3 - 6*x^2 + x + 1 = 0

Method 3 : Factor theory
  • Let y = F(x)
  • F(-1) = (-1)^6+ (-1)^5- 6*(-1)^4- 7*(-1)^3- 6*(-1)^2+ (-1)+ 1
  • F(-1) = 1 - 1 - 6 + 7 - 6 - 1 + 1 = -5
  • Remainder is -5, hence x = -1 is not a solution
  • F(+1) = (1)^6+ (1)^5- 6*(1)^4- 7*(1)^3- 6*(1)^2+ (1)+ 1
  • F(+1) = 1 + 1 - 6 - 7 - 6 + 1 + 1 = -15
  • Remainder is -15, hence x = 1 is not a solution

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Q04. Method 4 : Solve x^6 + x^5 - 6*x^4 - 7*x^3 - 6*x^2 + x + 1 = 0

Use equation theory
  • It has 4 real roots : q, r, s, t (From graph).
  • Then It has two complex roots : u = a + b*i and v = a - b*i
  • Sum of roots
    • q + r + s + t + u + v = -2
    • hence q + r + s + t + 2*a = -2 ... Eq (1)
  • Product roots
    • q*r*s*t*u*v = 1
    • Since u*v = a^2 + b^2
    • Hence q*r*s*t*(a^2+b^2) = 1 .................... Eq (2)
  • Assumption
    • Let q + r + s + t = 0 .......................... Eq (3)
    • From Eq (1), we have a = -1/2 .................. Eq (4)
    • Let q*r*s*t = 1 ................................ Eq (5)
    • From Eq (2), we have a^2 + b^2 = 1 ............. Eq (6)
    • From (4) and (6), we get a = -1/2 and b = Sqr(3)/2.
    • Hence complex roots are
      • u = a + b*i = -1/2 + i*Sqr(3)/2
      • u = a + b*i = -1/2 - i*Sqr(3)/2
  • Find q,r,s,t in Eq (3) and (5)
    • We also assume that r = -q, s = 1/q and t = -s
    • Find q = 0.381966 by using synthetic or factor theory.
    • Hence r = -0.381966, s = 2.618034 and t = -2.618034

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Q05. Method 5 : Solve x^6 + x^5 - 6*x^4 - 7*x^3 - 6*x^2 + x + 1 = 0

Assume that (x^2 + x + 1) is a factor of F(x)
  • ..... +1 +00 -07 -00 +1 (Coefficients of quotient)
  • --------------------------
  • +1 +1 -6 -07 -06 +01 +1 | 1 + 1 + 1
  • +1 +1 +1
  • ------------
  • +0 +0 -7 -07 -06 +01 +1 (Subtract above terms)
  • ..... -7 -07 -07
  • -------------
  • ...... 0 -00 +01 +01 +1 (Subtract above terms)
  • ............ +01 +01 +1
  • -------------------
  • ............. 0 +0 + 0 (Subtract above terms and Remainder is 0)
  • Hence (x^2 + x + 1) is a facto of F(x)
  • Hence F(x) = (x^2 + x + 1)*(x^4 -7*x^2 + 1)
  • Solution of F(x) = (x^2 + x + 1)*(x^4 - 7*x^2 +1) = 0
    • x^2 + x + 1 = 0 and x = (-1+Sqr(3))/2 and x = (-1-Sqr(3))/2
    • x^4 - 7*x^2 + 1 = 0
      • x^2 = (7 + Sqr(49 -4))/2 = 13.70820393/2 = 6.854101967
      • Hence x = 0.381966012 and x = -0.381966012
      • x^2 = (7 - Sqr(49 -4))/2 = 0.291796068/2 = 0.145898034
      • Hence x = 2.618033989 and x = -2.618033989

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Q06. Comments : Solve x^6 + x^5 - 6*x^4 - 7*x^3 - 6*x^2 + x + 1 = 0


  • We should know that (x^2 + x + 1) is a factor of the equation
  • Use graph of the equation on computer is fast to estimate the solutions

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Q07. Solve X^6 + 1 = 0
De Moivre's Theory : Draw a unit circle
  • 1. Draw 1st angle : A1 = (2*pi/6)/2 = 360/12
  • 2. Draw 2nd angle : A2 = 1*2*pi/6 + A1
  • 3. Draw 3nd angle : A3 = 2*2*pi/6 + A1
  • 4. Draw 4nd angle : A3 = 3*2*pi/6 + A1
  • 5. Draw 5nd angle : A3 = 4*2*pi/6 + A1
  • 6. Draw 6nd angle : A3 = 5*2*pi/6 + A1
Solutions : Use calculator
  • r1 = cos(A1) + i*sin(A1)
  • r2 = cos(A2) + i*sin(A2)
  • r3 = cos(A3) + i*sin(A3)
  • r4 = cos(A4) + i*sin(A4)
  • r5 = cos(A5) + i*sin(A5)
  • r6 = cos(A6) + i*sin(A6)
Solutions : By constuctions
  • Find (x1,y1) for angle A1. Measure x1 and y1
  • Similarly for other angles
  • Use the conjugate complex to reduce the calculation

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Q08. Solve X^7 - 1 = 0
De Moivre's Theory : Draw a unit circle
  • 1. Draw 1st angle : A1 = 1*2*pi/6
  • 2. Draw 2nd angle : A2 = 2*2*pi/6
  • 3. Draw 3nd angle : A3 = 3*2*pi/6
  • 4. Draw 4nd angle : A4 = 4*2*pi/6
  • 5. Draw 5nd angle : A5 = 5*2*pi/6
  • 6. Draw 6nd angle : A6 = 6*2*pi/6
Solutions : Use calculator
  • r1 = cos(A1) + i*sin(A1)
  • r2 = cos(A2) + i*sin(A2)
  • r3 = cos(A3) + i*sin(A3)
  • r4 = cos(A4) + i*sin(A4)
  • r5 = cos(A5) + i*sin(A5)
  • r6 = cos(A6) + i*sin(A6)
Solutions : By constuctions
  • Find (x1,y1) for angle A1. Measure x1 and y1
  • Similarly for other angles
  • Use the conjugate complex to reduce the calculation

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Q09. Answer

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Q10. Answer

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