What do the elements of $k[x]/((x − λ)^n$ look like? I want to show that ${1, [x − λ], ([x − λ])^2 , . . . , ([x − λ])^{n-1} }$ is a basis of $k[x]/((x − λ)^n$, with $λ \in k$.
But I don't understand what $k[x]/((x − λ)^n$ is.
What do the elements of $k[x]/((x − λ)^n$ look like?
Also what does $k[X]/(x^2+2x+3)$ look like?
 A: There is a canonical surjective ring homomorphism $\mathrm{pr}\colon k[x] → k[x]/(x - λ)^n$. So every element of $k[x]/(x - λ)^n$ can be written as $\operatorname{pr} (f)$ for some (non-unique) $f ∈ k[x]$.
I want to discuss the general situation. Whenever $R$ is a ring and $I$ is an ideal, there is a surjective ring homomorphism $\operatorname{pr}\colon R → R/I$ and so every element of $R/I$ can be written as $\operatorname{pr}(f)$ for some (non-unique) $f ∈ R$.
Some authors write $f + I$ for that element $\operatorname{pr}(f)$, others write $[f]_I$. Here, $f$ is a representative for $\operatorname{pr}(f)$.
Bottom line: They look exactly like elements of $k[x]$, except for some decoration (“$…+(x-λ)^n$” or “$[…]_{(x-λ)^n}$”) and you can calculate with them exactly as in $k[X]$ except equality holds way more often, so calculating becomes even easier.
Regarding your actual question:
Hint. Divide any representative $f$ of some element in $k[x]/(x-λ)^n$ with remainder by $(x-λ)^n$ to get another representative of the same element of degree less than $n$.
