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Mathematics Dictionary
Dr. K. G. Shih
Quartic Equations
Questions

Answers


Q01. Properties
  • 1. Equation : a*x^4 + b*x^3 + c*x^2 + d*x + e = 0
  • 2. Roots : It has four roots
    • Four real roots (all different or all same or two same)
    • Two real roots and two complex roots.
    • Complex roots are in pairs and they conjugate each other
  • 3. Conjugate : p + q*i is conjugate of p - q*i
    • Sum of conjugate is real e.g. (p+q*i) + (p-q*i) = 2*p
    • Product of conjugate is real e.g. (p+q*i)*(p-q*i) = p^2 + q^2
Example : What is conjugate of complex number ?
  • Conjugate of a + b*i is a - b*i.
  • Sum of complex number with it conjugate is 2*a (real)
  • Product of complex number with it conjugate is (a^2 + b^2)
  • Conjugate of complex and complex are symmetrical to x-axis.

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Q02. Method to solve

  • Factor theory.
  • Synthetic division.
  • Quartic formula method. Must be used if all roots are complex.

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Q03. Equation theory

  • Let p, q, r, s be the roots of a*x^4 + b*x^3 + c*x^2 + d*x + e = 0.
  • (x - p)*(x - q)*(x - r)*(x - q) = a*x^4 + b*x^3 + c*x^2 + d*x + e.
  • The coeff and roots relation
    • Sum of roots = p + q + r + s = -b/a.
    • Combination of 2 roots = p*q + p*r + p*s + q*r +q*s + rs = +c/a.
    • Combination of 3 roots = p*q*r + p*q*s + p*r*s + q*r*s = +d/a.
    • Combination of 4 roots = p*q*r*s = -e/a
Example : Equation x^4 + x^3 + x^2 + x + 1 = 0, find sum of roots.
  • Sum of roots = -(coeff of x^3)/(coeff of x^4) = -1.

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Q04. Solve x^4 + 1 = 0
General method
  • (x^2)^2 = -1.
  • x^2 = +i or x^2 = -i
  • For x^2 = i
    • x1 = Sqr(i) and x2 = -sqr(i)
  • For x^2 = -i
    • x3 = Sqr(-i) and x4 = - Sqr(-i)
  • How to find Sqr(i) ?
2nd method : use DeMoivre's theory
  • x^4 = -1 = cos(pi) + i*sin(pi)
  • x1 = cis((2*0*pi+pi)/4) = cis(1*pi/4) = cos(045) + i*sin(045).
  • x2 = cis((2*1*pi+pi)/4) = cis(3*pi/4) = cos(135) + i*sin(135).
  • x3 = cis((2*2*pi+pi)/4) = cis(5*pi/4) = cos(225) + i*sin(225).
  • x4 = cis((2*3*pi+pi)/4) = cis(7*pi/4) = cos(315) + i*sin(315).
  • Hence x1 = Sqr(i) = cos(45) + i*sin(45) = Sqr(2)*(1 + i)/2
  • Hence x2 = -Sqr(2)*(1 - i)/2 = conjugate of x1.
  • Hence x3 = Sqr(-i) = cos(225) + i*sin(225) = -Sqr(2)*(1 + i)/2
  • Hence x4 = -Sqr(2)*(1 - i)/2 = conjugate of x3.
3rd method : Construction method
  • Since X^4 = -1, we use angle pi/4 = 45 degrees
  • Draw a unit circle with center O and x-axis OX = 1
  • Draw point P on circle and angle POX = pi/4 = 45 degrees
  • Draw point Q on circle and angle QOX = 3*pi/4 = 135 degrees
  • Draw point R on circle and angle ROX = 5*pi/4 = 225 degrees
  • Draw point S on circle and angle SOX = 7*pi/4 = 315 degrees
  • Solution
    • x1 = cos(045) + i*sin(045) = Sqr(2)*(+1 + i)/2
    • x2 = cos(135) + i*sin(135) = Sqr(2)*(-1 + i)/2
    • x3 = cos(225) + i*sin(225) = Sqr(2)*(-1 - i)/2
    • x4 = cos(315) + i*sin(315) = Sqr(2)*(+1 - i)/2

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Q05. Solve x^4 - 1 = 0

General method
  • Use (x^4 - 1) = (x^2 + 1)*(x^2 -1) = 0
  • For x^2 + 1 = 0
    • x1 = +i and x2 = -i
  • For x^2 - 1 = 0
    • x3 = 1 and x4 = - 1
2nd method : use DeMoivre's theory
  • x^4 = 1 = cos(0) + i*sin(0)
  • x1 = cis((2*0*pi+0)/4) = cis(0*pi/4) = cos(000) + i*sin(000).
  • x2 = cis((2*1*pi+0)/4) = cis(2*pi/4) = cos(090) + i*sin(090).
  • x3 = cis((2*2*pi+0)/4) = cis(4*pi/4) = cos(180) + i*sin(180).
  • x4 = cis((2*3*pi+0)/4) = cis(6*pi/4) = cos(270) + i*sin(270).
  • Hence x1 = 1
  • Hence x2 = i
  • Hence x3 = -i
  • Hence x4 = -1
3rd method : Construction method
  • Since X^4 = 1, we use angle 0 degrees
  • Draw a unit circle with center O and x-axis OX = 1
  • Draw point P on circle and angle POX = 0 degrees
  • Draw point Q on circle and angle QOX = 90 degrees
  • Draw point R on circle and angle ROX = 180 degrees
  • Draw point S on circle and angle SOX = 270 degrees
  • Solution
    • x1 = cos(000) + i*sin(000) = 1
    • x2 = cos(090) + i*sin(090) = i
    • x3 = cos(180) + i*sin(180) = -1
    • x4 = cos(270) + i*sin(270) = -i

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Q06. Solve (x-5)*(x-7)*(x+6)*(x+4) = 504

Method
  • Expand and symplify to quartic equation
  • x^4 - 2*x^3 -61*x^2 + 62*x + 336 = 0
  • Use synthetic division, we get
  • (x-3)*(x-8)*(x+2)*(x+7) = 0
  • Hence roots are x=3, 8, -2 and -7.
Reference on internet
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Q07. F(x) = x^4 + x^3 + x^2 + x + 1

Show that F(1+i) = -4 + 5*i
  • Solve Equations
    Programs 4 and 6 : Find F(n) = ?
  • Show that F(1+i) = -4 + 5*i
    • Open the porgram and use option 6 to find F(n)
    • Give coeffcients : 1,1,1,1,1
    • Give real values 1, 1 for (1 + i)
    • We will get the answer
    • Give 0,0 to exit
  • Solve x^4 + x^3 + x^2 + x + 1 = 0
    • Open the porgram and use option 4 to solve equation
    • Give coeffcients : 1,1,1,1,1

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Q08. Answer

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Q09. Answer

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Q10. Answer

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