fattorizzazione:(1+x+x^2+x^3+x^5)^2 - x^5. come fattorizzo?
Calling Sₖ = (xᵏ - 1)/(x - 1), Sₚ = Φₚ(x) for prime p, and the given expression E = S₆² - x⁵ ⇒ (x - 1)²E = (x⁶ - 1)² - x⁵(x - 1)² = x¹² - 2x⁶ + 1 - x⁵(x² - 2x + 1) = x¹² - x⁷ - x⁵ + 1 = (x⁷ - 1)(x⁵ - 1) ⇒ E = S₇S₅ = Φ₇(x)Φ₅(x) = (x⁶ + x⁵ + x⁴ + x³ + x² + x + 1)(x⁴ + x³ + x² + x + 1).
From the diagram we deduce,
a) f(x) = 6 ⇒x = 18.
b) f(x) = 9 ⇒x = 36.
c) f(x) = 0 ⇒x = -18.
d) f(x) > 6 for x > 18.
e) f(x) ⩽ 9 for x ⩽ 36.
f) 0 < f(x) < 9 for -18 < x < 36.
(1 + x + x^2 + x^3 + x^4 + x^5)^2 - x^5
= (x^6 - 1)^2/(x - 1)^2 - x^5
= (x^12 - 2x^6 + 1 - x^7 + 2x^6 - x^5)/(x - 1)^2
= (x^12 - x^7 - x^5 + 1)/(x - 1)^2
= ((x^5 - 1)(x^7 - 1))/(x - 1)^2
= (x^4 + x^3 + x^2 + x + 1)(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)
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