# Thread: Express x⁴ + y⁴ + z⁴ - 2y²z² - 2z²x² - 2x²y² as the product of four factors.

1. ## Express x⁴ + y⁴ + z⁴ - 2y²z² - 2z²x² - 2x²y² as the product of four factors.

Hi,

How do I start this?

Express $x^4 + y^4 + z^4 - 2y^2z^2 - 2z^2x^2 - 2x^2y^2$ as the product of four factors.

2. ## Does this work?

If you put the numbers (x,y,z) = (2,1,1) your expression is zero. This means (x-y-z) could be a factor. By symmetry, (-x+y-z) and (-x-y+z) could also be factors. Last, (-2,1,1) also makes your expression zero. So (x+y+z) should be a factor. Try multiplying:
$
(x-y-z)(-x+y-z)(-x-y+z)(x+y+z)
$

3. Originally Posted by qmech
If you put the numbers (x,y,z) = (2,1,1) your expression is zero. This means (x-y-z) could be a factor. By symmetry, (-x+y-z) and (-x-y+z) could also be factors. Last, (-2,1,1) also makes your expression zero. So (x+y+z) should be a factor. Try multiplying:
$
(x-y-z)(-x+y-z)(-x-y+z)(x+y+z)
$
$(x-y-z)(-x+y-z)(-x-y+z)(x+y+z)$
$= x^4 + x^3z - y^2x^2 - x^2yz$

4. Am I corect?
Is the question finished?

5. $\left ( x-y-z \right )\left ( x+y-z \right )\left ( x-y+z \right )\left ( x+y+z \right )$
hope it's right !