# Coefficients of Product of Two Polynomials

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## Contents

## Theorem

Let $R$ be a commutative ring with unity.

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

For a natural number $n \ge 0$, 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 $f g$ is the sum:

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

### 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:

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

## Proof