Modulo Polynomial Division/Examples/Arbitrary Example 1

Example of Modulo Polynomial Division

Let:

\(\ds \map f x\)

\(=\)

\(\ds x^3 - x^2 - 1\)

\(\ds \map g x\)

\(=\)

\(\ds x + 1\)

Then $\map f x$ divided by $\map g x$ modulo $3$ is:

$x^2 + x - 1$

Proof

\(\ds \paren {x + 1} \paren {x^2 + x - 1}\)

\(=\)

\(\ds x^3 + 2 x^2 - 1\)

\(\ds \)

\(\equiv\)

\(\ds x^3 - x^2 - 1\)

\(\ds \pmod 3\)

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): division modulo $n$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): division modulo $n$