Question

Find the solution of the equation x³-2x²-3x in z₁₂

Answer 1

First let us factorize this polynomial :

$x^3-2x^2-3x=x\cdot(x-3)\cdot(x+1)$

The obvious solutions are $0; 3; 11$ (as $11=-1$ in $\mathbb{Z}_{12}$ ). However, as $\mathbb{Z}_{12}$ contains the zero divisors, there could possibly be other solutions.

$12=4\cdot 3$ and thus if $x$ is a solution of this equation, we should have $3|x$ or $3|(x+1)$ (as the condition $3|(x-3)$ is equivalent to $3|x$). Therefore, the other possible solutions may be $6;9;2;5;8$

Now we can also see, that if $x$ is even, then $x-3, x+1$ are odd and thus if $x$ is a solution and it is even, it should be divisible by 4. However, if $x$ is odd, then both $x-3, x+1$ are even, so their product is divisible by 4. These conditions yield that the solution set is

$S:= \{0;3;5;8;9;11 \}$