13. When is a polynomial f(x) of at most the order of a polynomial g(x) as x→∞? Give reasons for your answer.

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Understand the concept of the order of a polynomial as \( x \to \infty \). The order is determined by the highest power of \( x \) in the polynomial, which dominates the behavior of the polynomial for very large values of \( x \).

Let \( f(x) \) and \( g(x) \) be polynomials with degrees \( n \) and \( m \) respectively, where \( n \) and \( m \) are the highest powers of \( x \) in \( f(x) \) and \( g(x) \).

To say that \( f(x) \) is of at most the order of \( g(x) \) as \( x \to \infty \) means that the growth rate of \( f(x) \) does not exceed that of \( g(x) \). This happens if and only if \( n \leq m \).

This is because for large \( x \), the term with the highest power dominates, so if \( n > m \), then \( f(x) \) grows faster than \( g(x) \), and if \( n \leq m \), then \( f(x) \) grows at most as fast as \( g(x) \).

Therefore, the reason is that the degree of \( f(x) \) must be less than or equal to the degree of \( g(x) \) for \( f(x) \) to be of at most the order of \( g(x) \) as \( x \to \infty \).

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Polynomial Order and Degree

The order or degree of a polynomial is the highest power of the variable x with a nonzero coefficient. It determines the polynomial's growth rate as x approaches infinity, with higher-degree polynomials growing faster than lower-degree ones.

Asymptotic Behavior of Polynomials

As x approaches infinity, the term with the highest degree dominates the polynomial's behavior. Comparing two polynomials f(x) and g(x), the one with the higher degree grows faster, so f(x) is at most the order of g(x) if its degree is less than or equal to that of g(x).

Big-O Notation in Polynomial Comparison

Big-O notation describes an upper bound on the growth rate of functions. Saying f(x) is at most the order of g(x) means f(x) = O(g(x)) as x→∞, implying there exists a constant C such that |f(x)| ≤ C|g(x)| for sufficiently large x.