What is a polynomial regression?

$$\def\sa{a} \def\sb{b} \def\sc{c} \def\sd{d} \def\se{e} \def\sf{f} \def\sg{g} \def\sh{h} \def\si{i} \def\sj{j} \def\sk{k} \def\sl{l} \def\sm{m} \def\sn{n} \def\so{o} \def\sp{p} \def\sq{q} \def\sr{r} \def\ss{s} \def\st{t} \def\su{u} \def\sv{v} \def\sw{w} \def\sx{x} \def\sy{y} \def\sz{z} \def\va{\vec{a}} \def\vb{\vec{b}} \def\vc{\vec{c}} \def\vd{\vec{d}} \def\ve{\vec{e}} \def\vf{\vec{f}} \def\vg{\vec{g}} \def\vh{\vec{h}} \def\vi{\vec{i}} \def\vj{\vec{j}} \def\vk{\vec{k}} \def\vl{\vec{l}} \def\vm{\vec{m}} \def\vn{\vec{n}} \def\vo{\vec{o}} \def\vp{\vec{p}} \def\vq{\vec{q}} \def\vr{\vec{r}} \def\vs{\vec{s}} \def\vt{\vec{t}} \def\vu{\vec{u}} \def\vv{\vec{v}} \def\vw{\vec{w}} \def\vx{\vec{x}} \def\vy{\vec{y}} \def\vz{\vec{z}} \def\ga{\mathfrak{A}} \def\gb{\mathfrak{B}} \def\gc{\mathfrak{C}} \def\gd{\mathfrak{D}} \def\ge{\mathfrak{E}} \def\gf{\mathfrak{F}} \def\gg{\mathfrak{G}} \def\gh{\mathfrak{H}} \def\gi{\mathfrak{I}} \def\gj{\mathfrak{J}} \def\gk{\mathfrak{K}} \def\gl{\mathfrak{L}} \def\gm{\mathfrak{M}} \def\gn{\mathfrak{N}} \def\go{\mathfrak{O}} \def\gp{\mathfrak{P}} \def\gq{\mathfrak{Q}} \def\gr{\mathfrak{R}} \def\gs{\mathfrak{S}} \def\gt{\mathfrak{T}} \def\gu{\mathfrak{U}} \def\gv{\mathfrak{V}} \def\gw{\mathfrak{W}} \def\gx{\mathfrak{X}} \def\gy{\mathfrak{Y}} \def\gz{\mathfrak{Z}} \def\ra{A} \def\rb{B} \def\rc{C} \def\rd{D} \def\re{E} \def\rf{F} \def\rg{G} \def\rh{H} \def\ri{I} \def\rj{J} \def\rk{K} \def\rl{L} \def\rm{M} \def\rn{N} \def\ro{O} \def\rp{P} \def\rq{Q} \def\rr{R} \def\rs{S} \def\rt{T} \def\ru{U} \def\rv{V} \def\rw{W} \def\rx{X} \def\ry{Y} \def\rz{Z} \def\rva{\vec{A}} \def\rvb{\vec{B}} \def\rvc{\vec{C}} \def\rvd{\vec{D}} \def\rve{\vec{E}} \def\rvf{\vec{F}} \def\rvg{\vec{G}} \def\rvh{\vec{H}} \def\rvi{\vec{I}} \def\rvj{\vec{J}} \def\rvk{\vec{K}} \def\rvl{\vec{L}} \def\rvm{\vec{M}} \def\rvn{\vec{N}} \def\rvo{\vec{O}} \def\rvp{\vec{P}} \def\rvq{\vec{Q}} \def\rvr{\vec{R}} \def\rvs{\vec{S}} \def\rvt{\vec{T}} \def\rvu{\vec{U}} \def\rvv{\vec{V}} \def\rvw{\vec{W}} \def\rvx{\vec{X}} \def\rvy{\vec{Y}} \def\rvz{\vec{Z}} \def\seta{A} \def\setb{B} \def\setc{C} \def\setd{D} \def\sete{E} \def\setf{F} \def\setg{G} \def\seth{H} \def\seti{I} \def\setj{J} \def\setk{K} \def\setl{L} \def\setm{M} \def\setn{N} \def\seto{O} \def\setp{P} \def\setq{Q} \def\setr{R} \def\sets{S} \def\sett{T} \def\setu{U} \def\setv{V} \def\setw{W} \def\setx{X} \def\sety{Y} \def\setz{Z} \def\fa{a} \def\fb{b} \def\fc{c} \def\fd{d} \def\fe{e} \def\ff{f} \def\fg{g} \def\fh{h} \def\fi{i} \def\fj{j} \def\fk{k} \def\fl{l} \def\fm{m} \def\fn{n} \def\fo{o} \def\fp{p} \def\fq{q} \def\fr{r} \def\fs{s} \def\ft{t} \def\fu{u} \def\fv{v} \def\fw{w} \def\fx{x} \def\fy{y} \def\fz{z} \def\fA{A} \def\fB{B} \def\fC{C} \def\fD{D} \def\fE{E} \def\fF{F} \def\fG{G} \def\fH{H} \def\fI{I} \def\fJ{J} \def\fK{K} \def\fL{L} \def\fM{M} \def\fN{N} \def\fO{O} \def\fP{P} \def\fQ{Q} \def\fR{R} \def\fS{S} \def\fT{T} \def\fU{U} \def\fV{V} \def\fW{W} \def\fX{X} \def\fY{Y} \def\fZ{Z} \def\ma{A} \def\mb{B} \def\mc{C} \def\md{D} \def\me{E} \def\mf{F} \def\mg{G} \def\mh{H} \def\mi{I} \def\mj{J} \def\mk{K} \def\ml{L} \def\mm{M} \def\mn{N} \def\mo{O} \def\mp{P} \def\mq{Q} \def\mr{R} \def\ms{S} \def\mt{T} \def\matu{U} \def\mv{V} \def\mw{W} \def\mx{X} \def\my{Y} \def\mz{Z} \def\loss{\mathcal{L}} \newcommand{\dkl}[2]{D_{\text{KL}}\mathopen{}\paren{#1\,||\,#2}} \newcommand{\dataset}{S} \newcommand{\ndataset}{N} \newcommand{\idataset}{n} \newcommand{\inputRV}{\mathcal{X}} \newcommand{\inputvec}{\vec{x}} \newcommand{\ninputvec}[1]{\vec{x}_{#1}} \newcommand{\iinputvec}[1]{x_{#1}} \newcommand{\niinputvec}[2]{x_{#1, #2}} \newcommand{\icpnt}{i} \newcommand{\inputmatrix}{X} \newcommand{\inputdim}{D} \newcommand{\outputval}{y} \newcommand{\ioutputval}[1]{y_{#1}} \newcommand{\outputvec}{\vec{y}} \newcommand{\trainset}{S_{\text{train}}} \newcommand{\testset}{S_{\text{test}}} \newcommand{\truemodel}{f_{\text{true}}} \newcommand{\trainedmodel}{f_{\trainset}} \newcommand{\linmodel}[1]{f_{#1}} 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