Question

Find all n E N for which is the polynomial f(x) = x2n + x" + 1 divisible by g(x) = x2 + x + 1?​

Answer 1

Answer:

g(x) = x² + x + 1 = (x − ω)(x - ω²), where ω = (−1 + i√3)/2 is cube roots of unity. For cube roots of unity ω, we have, two important properties;

(i) ω³ = 1, (ii) (1+ω+ω²) = 0 . So in order that f(x) be divisible by g(x) , we must have f(ω) = 0 as well as f(ω²) = 0. That is;

f(ω) = ω^(2n) + ω^(n) + 1 as well as ,

f(ω²) = ω^(4n) + ω^(2n) + 1 must be 0. Now, if n is a multiple of 3, then the values above, are 3 in both the cases . But if n is not a multiple of 3 , the above values are 0 . Hence n should not be multiple of 3 .

Step-by-step explanation:

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