Solve 2(x)^(2)(y)^(2)+2(y)^(2)(z)^(2)+2(z)^(2)(x)^(2)-(x)^(4)-(y)^(4)-(z)^(4) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$2 { x }^{ 2 } { y }^{ 2 } +2 { y }^{ 2 } { z }^{ 2 } +2 { z }^{ 2 } { x }^{ 2 } - { x }^{ 4 } - { y }^{ 4 } - { z }^{ 4 }$$

Factor

$\left(x-y-z\right)\left(-x+y-z\right)\left(x+y-z\right)\left(x+y+z\right)$

Solution Steps

Rewrite $2x^{2}y^{2}+2y^{2}z^{2}+2z^{2}x^{2}-x^{4}-y^{4}-z^{4}$ as $\left(2yz\right)^{2}-\left(-x^{2}+y^{2}+z^{2}\right)^{2}$. The difference of squares can be factored using the rule: $a^{2}-b^{2}=\left(a-b\right)\left(a+b\right)$.

$$\left(2yz+x^{2}-y^{2}-z^{2}\right)\left(2yz-x^{2}+y^{2}+z^{2}\right)$$

Reorder the terms.

$$\left(x^{2}-y^{2}+2yz-z^{2}\right)\left(-x^{2}+y^{2}+2yz+z^{2}\right)$$

Consider $x^{2}-y^{2}+2yz-z^{2}$. Rewrite as difference of two terms.

$$x^{2}-\left(y^{2}-2yz+z^{2}\right)$$

Rewrite the terms as squares.

$$x^{2}-\left(y-z\right)^{2}$$

The difference of squares can be factored using the rule: $a^{2}-b^{2}=\left(a-b\right)\left(a+b\right)$.

$$\left(x-y+z\right)\left(x+y-z\right)$$

Consider $-x^{2}+y^{2}+2yz+z^{2}$. Rewrite $-x^{2}+y^{2}+2yz+z^{2}$ as $\left(y+z\right)^{2}-x^{2}$. The difference of squares can be factored using the rule: $a^{2}-b^{2}=\left(a-b\right)\left(a+b\right)$.

$$\left(y+z-x\right)\left(y+z+x\right)$$

Reorder the terms.

$$\left(-x+y+z\right)\left(x+y+z\right)$$

Rewrite the complete factored expression.

$$\left(-x+y+z\right)\left(x-y+z\right)\left(x+y-z\right)\left(x+y+z\right)$$

Evaluate

$\left(\left(x-z\right)^{2}-y^{2}\right)\left(-\left(x+z\right)^{2}+y^{2}\right)$