Motivation for studying rational curves Why do we study rational curves?
A curve $f(x,y)=0$ is called a rational curve if there exists two rational functions $\chi(t)$ and $\psi(t)$ such that $f(\chi(t),\psi(t))=0$ for all $t$.
Why is it not sufficient to study polynomials?
I read somewhere that we want to study conic sections but conic sections are not covered by polynomials. I do not really know what this means. Please enlighten me about this.
 A: 1) The curve  $f(x,y)=x^2+y^2-1=0$ (our friend the good old circle!) is a rational curve since it can be parametrized as $$x=\frac {t^2-1}{t^2+1} \quad ,\quad y=\frac {2t}{t^2+1}$$ 2) This is interesting because it allows one to find points on a circle with rational coordinates by plugging in rational numbers for $t$.
Equivalently it allows us to find right-angled triangles with integral sides.
For example: choosing $t=6$ we get $x=\frac {35}{37}, y= \frac {12}{37}$ so that $(\frac {35}{37})^2+(\frac {12}{37})^2=1$ and we get a right angle triangle with sides $35,12,37$ satisfying Pythagoras' relation $35^2+12^2=37^2$.
Amazingly,  the Babylonians somehow knew about this about 3800 years ago.   
3)  However it is impossible to find two polynomials (at least one of them non-constant)  $x=P(t), y=Q(t)$ satisfying $P^2(t)+Q^2(t)=1$.
Why? For example because the the limit of $P^2(t)+Q^2(t)$ for $t$ tending to infinity is infinity.       
4) So, yes, we need rational functions and not just polynomials  to define rational curves: mathematical terminology is sometimes pleasantly logical (or should I say rational ?)
