The problem before us was: show that the equations and have a root in common. I know that we asked you to show that they have a real root in common, but the method of solution will indeed demonstrate that every root of the cubic is also the root of the quintic. I will do this by hand, and then I will show you the MAPLE proof.
So let be any root of the cubic. Then . But then and . So
since was a root of the cubic. So is also a root of the quintic. C’est tout. In MAPLE you would do the following:
f1 := x^3-x-1:
f2 := x^5-x^4-1:
This is good, as this shows (why?) that the two polynomials have a root in common. Let us now find how many roots they have in common…
A := sylvester(f1,f2,x):
So the answer is three roots in common (why?)! Don’t tell me this is not neat. For now I will leave it to you to work out what exactly I did there — the code is strewn with clues. Please also explain to yourself why this is a proof.