Prove that the function fx 2x3 10x is Ox4SolutionWrite fx

Prove that the function f(x) = 2x^3 + 10x is O(x^4)

Solution

Write f(x) = p(x).q(x)

f(x) = 2x(x² + 5)

Let p(x) = x² + 5 and if we show p(x) is irreducible over Z₄, then f has O(x⁴)

Any polynomial p is irreducible over Z₄ if it does not have any roots in Z₄

p(-3) = 14 mod 4 = 2, p(-1) = 6 mod 4 = 2, p(1) = 2 and p(3) = 2.

Hence, p is irreducible over Z₄ or f is O(x⁴)