Real Polynomial Functions#

Definition: Real Polynomial Functions

A real polynomial function is a real function \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) for which there exists a real polynomial \(\sum_{k = 0}^n a_k x^k\) such that

\[ f(x) = \sum_{k = 0}^n a_k x^k = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0 \]

for every \(x \in \mathcal{D}\).

Theorem: Parity of Real Polynomial Functions

A real polynomial function \(f(x) = \sum_{k=0}^n a_k x^k\) is:

even if and only if only the coefficients \(a_k\) in front of even numbers \(k\) are non-zero;

Proof

TODO

Theorem: Continuity of Real Polynomial Functions

Every real polynomial function is continuous.

Proof

TODO

Theorem: Differentiability of Real Polynomial Functions

Every real polynomial function

\[ f(x) = \sum_{k=0}^n a_k x^k = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0 \]

is differentiable and its derivative \(f'\) is also a real polynomial function:

\[ f'(x) = \sum_{k = 1}^n k a_k x^{k-1} = n a_n x^{n-1} + (n-1)a_{n-1}x^{n - 2} + \cdots + 2a_2 x + a_1 \]

Proof

TODO

Theorem: Antidifferentiability of Real Polynomial Functions

Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) be a real function and let \(I \subseteq \mathcal{D}\) be an interval.

If \(f\) is polynomial on \(I\) with \(f(x) = \sum_{k=0}^n a_k x^k\), then \(f\) is antidifferentiable on \(I\) with

\[ \int f(x) \mathop{\mathrm{d}x} = C + \sum_{k=0}^n \frac{1}{k+1}a_k x^{k+1}. \]

Proof

TODO