Functions

• $\mathcal{F}(S,F)$ denotes all functions $f$ where $f:S\to F$
• Definitions ($f,g\in \mathcal{F}(S,F)$)
  • $f$ and $g$ are equal if $f(s)=g(s),\forall s\in S$
  • $\mathcal{F}(S,F)$ is a vector space with $(f+g)(s)=f(s)+g(s)$ and $(cf)(s)=c[f(s)]$

Polynomials

• $f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+a_{1}x+a_0$, each $a_k\in F$ (degree $n$)
• If $a_n=a_{n-1}=\cdots =a_0=0$ then $f$ is the zero polynomial (degree -1)
• Two polynomials of degree $n$ are equal if all coefficients are equal

Vector Spaces

• The set of all polynomials of field F and degree $n$ form a vector space $P_n(F)$
• Let $f(x)=a_n{x}^n+\cdots +a_{1}x+a_0$, $g(x)=b_m{x}^m+\cdots +b_{1}x+b_0$ where $m\le n$
• Define $b_{m+1}=b_{m+2}=\cdots b_n=0$ (make $g$ degree n by padding with zero coefficients)
• Define
  • $f(x)+g(x)=(a_n+b_n)x^n+(a_{n-1}+b_{n-1})x^{n-1}+\cdots +(a_1+b_1)x+a_0+b_0$
  • For any $c\in F$, $cf(x)=c{a}_n{x}^n+\cdots +c{a}_{1}x+c{a}_0$