Factor the following expression:

$$
A=x^{3}(y-z)+y^{3}(z-x)+z^{3}(x-y)
$$
I. Solution: If we convert the second and third terms into polynomials and regroup the terms, we notice that $(y-z)$ can be factored out:

$$
\begin{aligned}
A & =x^{3}(y-z)+y^{3} z-y^{3} x+z^{3} x-z^{3} y= \\
& =x^{3}(y-z)-x\left(y^{3}-z^{3}\right)+y z\left(y^{2}-z^{2}\right)= \\
& =(y-z)\left[x^{3}-x\left(y^{2}+y z+z^{2}\right)+y z(y-z)\right]= \\
& =(y-z)\left(x^{3}-x y^{2}-x y z-x z^{2}+y^{2} z+y z^{2}\right)
\end{aligned}
$$

Since $A$ is symmetric in $x, y, z$ (i.e., $A$ remains unchanged under cyclic permutations of $x, y, z$), $(z-x)$ and $(x-y)$ must also be factorable. Group the terms in the second factor appropriately:

$$
\begin{aligned}
A & =(y-z)\left[x\left(x^{2}-y^{2}\right)-y z(x-y)-z^{2}(x-y)\right]= \\
& =(y-z)(x-y)\left(x^{2}+x y-y z-z^{2}\right)= \\
& =(y-z)(x-y)\left[x^{2}-z^{2}+y(x-z)\right]= \\
& =(y-z)(x-y)(x-z)(x+z+y)=-(x-y)(y-z)(z-x)(x+y+z)
\end{aligned}
$$

Klara Madarász (Szeged, Tömörkény Secondary School, 2nd grade)

II. Solution: $A$ can be considered as a third-degree polynomial in $x$. If we denote the roots of the equation $A=0$ by $x_{1}, x_{2}, x_{3}$, then as is well known,

$$
A=\left(x-x_{1}\right)\left(x-x_{2}\right)\left(x-x_{3}\right)
$$

where $a$ is a constant independent of $x$.

Notice that if we substitute $y$ or $z$ for $x$ in $A$, $A$ becomes 0, so $x_{1}=y$ and $x_{2}=z$, and thus the determination of $a$ and $x_{3}$ remains from the following equation:

$$
A=x^{3}(y-z)+y^{3}(z-x)+z^{3}(x-y)=a(x-y)(x-z)\left(x-x_{3}\right)
$$

The coefficients of the same powers of $x$ on the left and right sides are equal.

The coefficient of $x^{3}$ on the left side is $(y-z)$, and on the right side it is $a$, so

$$
a=y-z
$$

The coefficient of $x^{2}$ on the left side is 0, and on the right side it is $-a y-a z-a x_{3}$, so

$$
-a y-a z-a x_{3}=0
$$

from which

$$
x_{3}=-(y+z)
$$

Therefore,

$$
A=(y-z)(x-y)(x-z)(x+y+z)=-(x-y)(y-z)(z-x)(x+y+z)
$$

Mária Jáky (Pécs, Bányaip. Secondary School, 3rd grade)