Polynomial Divisor Modulo Integer/Examples/Arbitrary Example 1
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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 $n$