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Mathematics Dictionary
Dr. K. G. Shih
Power 7 Equation
Questions

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  • Q01 | - Method 1 : Solve x^7+ 2*x^6- 5*x^5- 13*x^4- 13*x^3- 5*x^2+ 2*x+ 1= 0
  • Q02 | - Method 2 : Solve x^7+ 2*x^6- 5*x^5- 13*x^4- 13*x^3- 5*x^2+ 2*x+ 1= 0
  • Q03 | - Method 3 : Solve x^7+ 2*x^6- 5*x^5- 13*x^4- 13*x^3- 5*x^2+ 2*x+ 1= 0
  • Q04 | - Method 4 : Solve x^7+ 2*x^6- 5*x^5- 13*x^4- 13*x^3- 5*x^2+ 2*x+ 1= 0
  • Q05 | - Method 5 : Solve x^7+ 2*x^6- 5*x^5- 13*x^4- 13*x^3- 5*x^2+ 2*x+ 1= 0
  • Q06 | - Comments : Solve x^7+ 2*x^6- 5*x^5- 13*x^4- 13*x^3- 5*x^2+ 2*x+ 1= 0
  • Q07 | - Solve x^7 + 1 = 0
  • Q08 | - Solve x^7 - 1 = 0
  • Q09 | -
  • Q10 | -
Answers


Q01. Solve x^7 + 2*x^6 - 5*x^5 - 13*x^4 - 13*x^3 - 5*x^2 + 2*x + 1 = 0 Method 1 : Graphic solution
  • Study Program | Graphic Solutions of Polynomial Functions.
  • Use program 01 11, we get the following real roots by estimation
    • -1, (or by synthetic division)
    • 0.382, -0.382, 2,618 and -2.618 (Approx graphic solutions of program 01 11)

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Q02. Solve x^7 + 2*x^6 - 5*x^5 - 13*x^4 - 13*x^3 - 5*x^2 + 2*x + 1 = 0

Method 2 : Synthetic Division
  • 1 +2 -5 -13 -13 -5 +2 +1 | -1
  • . -1 -1 +06 +07 +6 -1 -1
  • --------------------------
  • 1 +1 -6 -07 -06 +1 +1 00
  • Hence x = -1 is a solution
  • But other real solution is not easy to find.
  • But we now that this equation is (x+1)*(x^6+x^5-6*x^4-7*x^3-6*x^2+x+1 = 0

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Q03. Solve x^7 + 2*x^6 - 5*x^5 - 13*x^4 - 13*x^3 - 5*x^2 + 2*x + 1 = 0

Method 3 : Factor theory
  • Let y = F(x)
  • F(-1) = (-1)^7+ 2*(-1)^6- 5*(-1)^5- 13*(-1)^4- 13*(-1)^3- 5*(-1)^2+ 2*(-1)+ 1
  • F(-1) = -1 + 2 + 5 - 13 + 13 - 5 - 2 + 1 = 0
  • Hence x = -1 is a solution
  • But other real solution is not easy to find.
  • Use sythetic division, we can get factor form, bur this method we can not.
  • Not : Use Program 45 in the program of graphic solution
    • Let u = 0.381966011250205
    • F(+u) = 0.00000
    • F(-u) = 0.00000
    • F(+1/u) = 0.00000
    • F(-1/u) = 0.00000
    • Real solutions : 0.381966, -0.381966, 2.618 and -2.618
    • Other two complex roots have to find

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Q04. Solve x^7 + 2*x^6 - 5*x^5 - 13*x^4 - 13*x^3 - 5*x^2 + 2*x + 1 = 0

Method 4 : Use equation theory
  • It has 5 real roots : p, q, r, s, t (From graph).
  • Then It has two complex roots : u = a + b*i and v = a - b*i
  • Sum of roots
    • p + q + r + s + t + u + v = -2
    • Since p = -1, hence q + r + s + t + 2*a = -1 ... Eq (1)
  • Product roots
    • p*q*r*s*t*u*v = -1
    • Since p = -1 and 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. Solve x^7 + 2*x^6 - 5*x^5 - 13*x^4 - 13*x^3 - 5*x^2 + 2*x + 1 = 0

Method 5 : Assume that (x^2 + x + 1) is a factor of F(x)
  • ..... +1 +01 -07 -07 +1 +1 (Coefficients of quotient)
  • --------------------------
  • +1 +2 -5 -13 -13 -05 +2 +1 | 1 + 1 + 1
  • +1 +1 +1
  • ------------
  • +0 +1 -6 -13 -13 -05 +2 + 1 (Subtract above terms)
  • .. +1 +1 +01
  • -------------
  • ... 0 -7 -14 -13 -05 +2 + 1 (Subtract above terms)
  • ..... -7 -07 -07
  • -------------------
  • ...... 0 -07 -06 -05 +2 + 1 (Subtract above terms)
  • ......... -7 -07 -07
  • -----------------------
  • .......... 0 +01 +02 +2 + 1 (Subtract above terms)
  • ............. +1 +1 +1
  • -----------------------
  • .............. 0 1 +1 + 1 (Subtract above terms)
  • .................. 1 +1 + 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^5 + x^4 -7*x^3 - 7*x^2 +x + 1)
  • Prove that (x+1) is a factor of x^5 + x^4 -7*x^3 - 7*x^2 +x + 1
    • +1 +1 -07 -07 +01 +01 | -1
    • .. -1 -00 +07 -00 -01
    • ------------------------
    • +1 +0 -07 +00 +01 +00 (Add above terms and remainder is 0)
    • Hence (x+1) is factor
    • Hence F(x) = (x + 1)*(x^2 + x + 1)*(x^4 - 7*x^2 +1) = 0
  • Solution of F(x) = (x + 1)*(x^2 + x + 1)*(x^4 - 7*x^2 +1) = 0
    • x + 1 = 0 and x = -1
    • 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
  • We should know that (x+1) and (x^2 + x + 1) are factors of the equation
  • Use graph of the equation on computer is fast to estimate the solutions

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Q07. Solve X^7 + 1 = 0

De Moivre's Theory : Draw a unit circle
  • 1. Draw 1st point on circle with angle : A1 = (2*pi/7)/2 = 360/14
  • 2. Draw 2nd point on circle with angle : A2 = 1*2*pi/7 + A1
  • 3. Draw 3nd point on circle with angle : A3 = 2*2*pi/7 + A1
  • 4. Draw 4nd point on circle with angle : A3 = 3*2*pi/7 + A1
  • 5. Draw 5nd point on circle with angle : A3 = 4*2*pi/7 + A1
  • 6. Draw 6nd point on circle with angle : A3 = 5*2*pi/7 + A1
  • 7. Draw 7nd point on circle with angle : A3 = 6*2*pi/7 + 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)
  • r7 = cos(A7) + i*sin(A7)
Solutions : By constuctions
  • For angle A1 : r1 = x1 + i*y1. Measure x1 and y1
  • For angle A2 : r2 = x2 + i*y2. Measure x2 and y2
  • Similarly find other five solutions
  • 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 = 0*2*pi/7 = 000
  • 2. Draw 2nd angle : A2 = 1*2*pi/7 = 051.428 degrees
  • 3. Draw 3nd angle : A3 = 2*2*pi/7 = 102.857 degrees
  • 4. Draw 4nd angle : A4 = 3*2*pi/7 = 154.285 degrees
  • 5. Draw 5nd angle : A5 = 4*2*pi/7 = 205.714 degrees
  • 6. Draw 6nd angle : A6 = 5*2*pi/7 = 257.142 degrees
  • 7. Draw 7nd angle : A7 = 6*2*pi/7 = 308.570 degrees
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)
  • r7 = cos(A7) + i*sin(A7)
Solutions : By constuctions
  • For angle A1 : r1 = x1 + i*y1 and find x1,y1
  • For angle A2 : r2 = x2 + i*y2 and find x2,y2
  • Similarly for other 5 roots
  • Use the conjugate complex to reduce the calculation
Answer

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Q09. Answer

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Q10. Answer

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