Polynomial Divisor Modulo Integer/Examples/Arbitrary Example 1

Examples of Polynomial Divisors Modulo $m$

Let $\map f x$ be the polynomial:

$\map f x = 2 x^4 - 4 x - 3$

Then $\map f x$ has the following (polynomial) divisors modulo $7$:

$2 x^2 + 3 x + 3$

$x - 2$

$x + 4$

Proof

\(\ds \paren {2 x^2 + 3 x + 3} \paren {x - 2} \paren {x + 4}\)

\(=\)

\(\ds 2 x^4 + 7 x^3 - 7 x^2 - 18 x - 24\)

\(\ds \)

\(=\)

\(\ds 2 x^4 + 7 x^3 - 7 x^2 - \paren {2 \times 7 + 4} x - \paren {3 \times 7 + 3}\)

\(\ds \)

\(\equiv\)

\(\ds 2 x^4 + 0 x^3 - 0 x^2 - 4 x - 3\)

\(\ds \pmod 7\)

\(\ds \)

\(\equiv\)

\(\ds 2 x^4 - 4 x - 3\)

\(\ds \pmod 7\)

$\blacksquare$

Sources

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