Problem:
Solution:
Because the polynomial with a single variable is a principal ideal domain. The ideal ⟨x3+x2−4x−4,x3−x2−4x+4,x3−2x2−x+2⟩ is essentially generated by their GCD, so we find the GCD of these polynomials using the program we developed for the last problem, it turns out to be ⟨x−2⟩.
Now it is obvious that x2−4=(x+2)(x−2)∈⟨x−2⟩=⟨x3+x2−4x−4,x3−x2−4x+4,x3−2x2−x+2⟩.
Solution:
Because the polynomial with a single variable is a principal ideal domain. The ideal ⟨x3+x2−4x−4,x3−x2−4x+4,x3−2x2−x+2⟩ is essentially generated by their GCD, so we find the GCD of these polynomials using the program we developed for the last problem, it turns out to be ⟨x−2⟩.
Now it is obvious that x2−4=(x+2)(x−2)∈⟨x−2⟩=⟨x3+x2−4x−4,x3−x2−4x+4,x3−2x2−x+2⟩.
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