Exercise 1.5.3

Question

Prove that the form $f(X,Y) = 6X^2 - 5XY - 6Y^2$ is reducible in $\Z[X,Y]$ and determine its factorizations.

Richard Ganaye · Accepted Answer

\begin{proof}
The discriminant  $\Delta = 169 = 13^2$ of $f$ is a square, thus $f$ is reducible.

Indeed,
\begin{align*}
24 f(X,Y) &= (12 X - 5 Y)^2 - 13^2 Y^2\
&= (12 X - 18Y)(12 X + 8 Y),
\end{align*}
so
$$f(X,Y) = (2X - 3Y)(3X + 2Y).$$
\end{proof}

Exercise 1.5.3

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Exercise 1.5.3

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