Let $p(x)=x^4+ax^3+bx^2+cx+d$ (where $a,b,c,d\in\mathbb{R}$) and $p(1)=10,p(2)=20,p(3)=30$. How can I find the quartic $p(x)$ through the info given?
I know how to start that is firstly considering an equation like $y=(x-1)(x-2)(x-3)+ax^4$ but after that it gets a bit confusing as we have to add some specific powers of $x$ with coefficient like $ax^3+bx+d$ on RHS. So my question is how to know which powers of $x$ we have to add in the above equation, how to know that other wise if we add $ax^4+bx^3+cx^2+dx+e$ then there won't be any use of considering y like I did in the starting. Moreover it becomes quite lengthy if we add more and more terms. How to deal with it?
PS~I have not added the whole question, but written everything I have to ask.