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Figure 27 : Example to Solve quaint equation F(x)=0
Product no. Program 17 05
Q1. This diagram can be used to solve
    1. x^5 + 3*x^4 - 5*x^3 - 15*x^2 + 4*x + 12 = 0
    2. x^5 + 3*x^4 - 5*x^3 - 15*x^2 + 4*x + 12 > 0. See Q2
    3. x^5 + 3*x^4 - 5*x^3 - 15*x^2 + 4*x + 12 < 0. See Q3
Q2. Solve y = x^5 + 3*x^4 - 5*x^3 - 15*x^2 + 4*x + 12 > 0
A2. From diagram we have y = (x+3)*(x+2)*(x+1)*(x-1)*(x-2)
  • If x < -3,
    • x+3=(-), (x+2)=(-), (x+1)=(-), (x-1)=(-), (x-2)=(-)
    • Hence y < 0 when x < -3 is not an answer
  • if -3 < x < -2,
    • x+3=(+), (x+2)=(-), (x+1)=(-), (x-1)=(-), (x-2)=(-)
    • Hence y > 0 when -3 < x < -2 is an answer
  • if -2 < x < -1,
    • x+3=(+), (x+2)=(+), (x+1)=(-), (x-1)=(-), (x-2)=(-)
    • Hence y < 0 when -2 < x < -1 is not an answer
  • if -1 < x < 1,
    • x+3=(+), (x+2)=(+), (x+1)=(+), (x-1)=(-), (x-2)=(-)
    • Hence y > 0 when -1 < x < 1 is an answer
  • if 1 < x < 2,
    • x+3=(+), (x+2)=(+), (x+1)=(+), (x-1)=(+), (x-2)=(-)
    • Hence y < 0 when 1 < x < 2 is not an answer
  • if x > 2,
    • x+3=(+), (x+2)=(+), (x+1)=(+), (x-1)=(+), (x-2)=(+)
    • Hence y > 0 when x > 2 is an answer
Q3. Solve y = x^5 + 3*x^4 - 5*x^3 - 15*x^2 + 4*x + 12 < 0
A3. From diagram we have y = F(x) = (x+3)*(x+2)*(x+1)*(x-1)*(x-2)
    • If x < -3,
      • x+3=(-), (x+2)=(-), (x+1)=(-), (x-1)=(-), (x-2)=(-)
      • Hence y < 0 when x < -3 is an answer
    • if -3 < x < -2,
      • x+3=(+), (x+2)=(-), (x+1)=(-), (x-1)=(-), (x-2)=(-)
      • Hence y > 0 when -3 < x < -2 is not an answer
    • if -2 < x < -1,
      • x+3=(+), (x+2)=(+), (x+1)=(-), (x-1)=(-), (x-2)=(-)
      • Hence y < 0 when -2 < x < -1 is an answer
    • if -1 < x < 1,
      • x+3=(+), (x+2)=(+), (x+1)=(+), (x-1)=(-), (x-2)=(-)
      • Hence y > 0 when -1 < x < 1 is not an answer
    • if 1 < x < 2,
      • x+3=(+), (x+2)=(+), (x+1)=(+), (x-1)=(+), (x-2)=(-)
      • Hence y < 0 when 1 < x < 2 is an answer
    • if x > 2,
      • x+3=(+), (x+2)=(+), (x+1)=(+), (x-1)=(+), (x-2)=(+)
      • Hence y > 0 when x > 2 is not an answer

    We can also write as below to find F(x) < 0

      ...... -3 < x .. -3 < x < -2 .. -2 < x < -1 .. -1 < x < 1 .. 1 < x < 2 .. X > 2
      (x+3) ... - ......... + ............ + ............ + .......... + ........ +
      (x+2) ... - ......... - ............ + ............ + .......... + ........ +
      (x+1) ... - ......... - ............ - ............ + .......... + ........ +
      (x-1) ... - ......... - ............ - ............ - .......... + ........ +
      (x-2) ... - ......... - ............ - ............ - .......... - ........ +
      F(x) .... - ......... + ............ - ............ + .......... - ........ +
Q4. Find range of y = x^5 + 3*x^4 - 5*x^3 - 15*x^2 + 4*x + 12
A4. Fow following domain
    1. When x is between -2 and -1
    • The minimum point at x=-1.5022 and y=-3.25
    • Hence the range is from -3.25 < y < 0
    2. When x is between 1 and 3
    • The minimum point at x=1.605 and y=-10.335
    • Hence the range is from -10.335 < y < 0
Q5. Find F(-1.5022)
  • Use MD2002 ZM17 program 09
  • Give coefficients : 1, 3, -5, -15, 4, 12
  • Give root a+b*i : -1.6022, 0
Q6. Find F(1.605)
  • Use MD2002 ZM17 program 09
  • Give coefficients : 1, 3, -5, -15, 4, 12 (not required because given in Q5)
  • Give root a+b*i : -1.605, 0
Q6. Reference
  • Solve quaint equation : MD2002 ZM17 program 05
  • Find F(a+b*i) : MD2002 ZM17 program 09
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