Appendix
A-10: Solving Equations
Example A-10.4
Iffx=x7+3x4+2 x−1, solve the equation fx=0.
Solution
Interactive Solution
The graph of fx in Figure A-10.4(a) suggests the equation fx=0 has just one real root, and therefore three pairs of complex-conjugate roots.
In general, a seventh-degree polynomial will not have solutions expressible as radicals, and even if it did, the expressions would most likely be very cumbersome.
Hence, the expectation here is a numeric solution via the Solve≻Numerically Solve (w/complex) option in the Context Panel.
P:=x^7+3*x^4+2*x-1:
plot(P,x=-1.5..1,y=-3..2);
Figure A-10.4(a) Graph of fx
Control-drag the polynomial and press the Enter key.
Context Panel: Solve≻Numerically Solve (w/complex)
x7+3x4+2 x−1
x7+3x4+2x−1
→solve
−1.19259593419359−0.179308039007199I,−1.19259593419359+0.179308039007199I,0.245204803152949−0.898051588497434I,0.245204803152949+0.898051588497434I,0.441417709017076,0.726682276532101−1.12661189290359I,0.726682276532101+1.12661189290359I
It certainly would be nice if Maple would display the roots as per Table A-10.4(a).
0.441417709017076
−1.19259593419359 ±0.179308039007199 i
0.245204803152949 ±0.898051588497434 i
0.726682276532101 ±1.12661189290359 i
Table A-10.4(a) Roots of the equation fx=0
The solution given by the Solve≻Numerically Solve (w/complex) option in the Context Panel is the simplest interactive solution. The alternatives below are far more tedious to implement and are not really recommended in this context.
The Context Panel option Solve≻Numerically Solve provides just the real solution, x=0.4414177090.
The Context Panel option Solve≻Obtain Solutions for≻x returns all seven solutions as a sequence of RootOf structures, of the form
RootOf_Z7+3_Z4+2_Z−1,index=k
where k ranges from 1 to 7. Each member of this sequence has to be extracted to a separate line (easiest by Control-drag or copy/paste) so that the Approximate option in the Context Panel can be applied. The results for k=2,…,7, are the complex conjugate pairs listed in Table A-10.4(a).
Coded Solution
Assign the polynomial to the name f.
f≔x7+3x4+2 x−1
f:=x7+3x4+2x−1
Apply the fsolve command with the "complex" option. This gives a numeric approximation to all solutions of a polynomial equation.
fsolvef,x,complex
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