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 $f^{\prime }$ is also a real polynomial function:

$$f^{\prime }(x)=\sum _{k=1}^{n}k{a}_k{x}^{k-1}=n{a}_n{x}^{n-1}+(n-1)a_{n-1}{x}^{n-2}+\cdots +2a_{2}x+a_1$$

PROOF

TODO

Theorem: Antidifferentiability of Real Polynomial Functions

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 antidifferentiable and its antiderivatives $\int f(x)\mathrm{dx}$ are also real polynomial functions:

$$\int f(x)\mathrm{dx}=\sum _{k=0}^n\frac{1}{k+1}{a}_k{x}^{k+1}+C=\frac{1}{n+1}{a}_n{x}^{n+1}+\frac{1}{n}{a}_{n-1}{x}^n+\cdots +\frac{1}{2}{a}_1{x}^2+a_{0}x+C$$

PROOF

TODO