Application of mathematical induction If $p (x)= a_0 + a_1x^2+....a_nx^n$ in $\Bbb R [x]$ and $a \in \Bbb R$,  then $p (x)$ can be written as $b_0+b_1(x-a)+.....b_n(x-a)^n$ ,Where $b_i \in\Bbb R \forall i\in\{0,1,2\ldots n\}$. 
Confusion: IF two polynomials are equal they both must have the same degree and the coefficients of the corresponding unknowns with equal powers are also equal. the polynomial $f(x) = x^3 + 2x^2 + 5$ is basically the structure $( )^3 + 2( )^2 + 5$ and not what we put into those parenthesis. Then how this problem is meaningful . I must be going horribly wrong some where. Please help.
* Why can't I type mathematical symbols? I am following the codes in the link : MathJax basic tutorial and quick reference 
 A: Your definition is correct in a sense. In algebra one often defines a polynomial to be just the structure you mentioned. (I.e. $x$ is an "indeterminate", not a real variable.)
But this question is not asking about polynomials defined this way. When the question writes "$p(x)$ can be written as $q(x-a)$", it's not referring to $x-a$ as an indeterminate, in the way that you're thinking. It's asking, in the sense of functions on $\mathbb{R}$, for you to show there are coefficients on $q$ such that the functions $p(x)$ and $q(x-a)$ are everywhere equal.
You should be able to do the base case. Suppose it's true for degree $n-1$. Then for degree $n$, we have $$a_nx^n + \dotsb + a_1x+a_0 \\[12pt] =a_n(x-a)^n + (\text{Stuff of degree $<n$})+a_{n-1}x^{n-1}+\dotsb + a_0.$$
Can you find which terms to use induction on?
To type math code, your syntax should look like
$p(x) = a_0 + a_1x + \cdots b + a_nx^n$

gives $p(x) = a_0 + a_1x + \dotsb + a_nx^n$.
Comment: I would prefer to prove this a different way: given $p(x) = a_nx^n + \dotsb + a_0$, let $\zeta + a=x$, and expand $p(\zeta + a)$. Then substitute $\zeta = x-a$.
A: Two polynomials are called equal, if they yield the same value for any x.
