Coefficients of Product of Two Polynomials

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Theorem

Let $R$ be a commutative ring with unity.

Let $f, g \in R[x]$ be polynomials over $R$.

For a natural number $n \geq 0$, let:

$a_n$ be the coefficient of the monomial $x^n$ in $f$.
$b_n$ be the coefficient of the monomial $x^n$ in $g$.


As an indexed summation

The coefficient $c_n$ of $x^n$ in $fg$ is the sum:

$c_n = \displaystyle \sum_{k \mathop = 0}^n a_k b_{n-k}$


As an indexed summation bounded by degrees

Let $\deg f$ and $\deg g$ be their degrees.


The coefficient $c_n$ of $x^n$ in $fg$ is the sum:

$c_n = \displaystyle \sum_{k \mathop = n - \deg g}^{\deg f} a_k b_{n-k}$


Proof


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