The factors of `x(y^2-z^2)+y(z^2-x^2)+z(x^2-y^2)` are:

Options

A. (x-y)(y-z)(z-x)
B. (x+y)(y+z)(z+x)
C. (y-x)(z-y)(x-z)
D. (x+y)(z-y)(x-z)

Answer

To factor the expression \( x(y^2 - z^2) + y(z^2 - x^2) + z(x^2 - y^2) \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ x(y^2 - z^2) + y(z^2 - x^2) + z(x^2 - y^2) \]
We can use the difference of squares formula \( a^2 - b^2 = (a - b)(a + b) \) to rewrite the terms:
\[ = x(y - z)(y + z) + y(z - x)(z + x) + z(x - y)(x + y) \]

### Step 2: Expand the expression
Next, we will expand each term:
\[ = xy^2 - xz^2 + yz^2 - yx^2 + zx^2 - zy^2 \]
Rearranging gives:
\[ = xy^2 - zy^2 + yz^2 - yx^2 + zx^2 - xz^2 \]

### Step 3: Group the terms
Now, we can group the terms:
\[ = (xy^2 - zy^2) + (yz^2 - yx^2) + (zx^2 - xz^2) \]
Factoring out common terms from each group:
\[ = y^2(x - z) + z^2(y - x) + x^2(z - y) \]

### Step 4: Factor out common factors
Now we can factor out \( (x - y)(y - z)(z - x) \):
\[ = (x - y)(y - z)(z - x) \]

### Final Expression
Thus, the factors of the expression \( x(y^2 - z^2) + y(z^2 - x^2) + z(x^2 - y^2) \) are:
\[ (x - y)(y - z)(z - x) \]