Question

Factorizar la siguiente expresión porfa :,v
$P(x,y,z)=x^{4}(y-z)+y^{4}(z-x)+z^{4}(x-y)$

Answer 1

Respuesta:

(y-z)(y-x)(z-x)(x²+y²+z²+zx+yz+yx)

Explicación paso a paso:

*se sabe:

a³-b³=(a-b)(a²+b²+ab)

a²-b²=(a-b)(a+b)

*factorizando:

P(x,y,z)=x⁴(y-z)+y⁴(z-x)+z⁴(x-y)

P(x,y,z)=x⁴(y-z)+y⁴z-y⁴x+z⁴x-z⁴y

P(x,y,z)=x⁴(y-z)-y⁴x+z⁴x+y⁴z-z⁴y

P(x,y,z)=x⁴(y-z)-x(y⁴-z⁴)+yz(y³-z³)

P(x,y,z)=x⁴(y-z)-x(y²-z²)(y²+z²)+yz(y-z)(y²+z²+yz)

P(x,y,z)=x⁴(y-z)-x(y-z)(y+z)(y²+z²)+yz(y-z)(y²+z²+yz)

P(x,y,z)=(y-z)[ x⁴-x(y+z)(y²+z²)+yz(y²+z²+yz)]

P(x,y,z)=(y-z)[x⁴-(xy+xz)(y²+z²)+y³z+yz³+y²z²]

P(x,y,z)=(y-z)[x⁴-(xy³+xyz²+xzy²+xz³)+y³z+yz³+y²z²]

P(x,y,z)=(y-z)[x⁴-xy³-xyz²-xzy²-xz³+y³z+yz³+y²z²]

P(x,y,z)=(y-z)[x⁴-xy³-xzy²+y³z-xz³+yz³-xyz²+y²z²]

P(x,y,z)=(y-z)[x(x³-y³)+y²z(y-x)+z³(y-x)+yz²(y-x)]

P(x,y,z)=(y-z)[x(x-y)(x²+y²+xy)+y²z(y-x)+z³(y-x)+yz²(y-x)]

P(x,y,z)=(y-z)[-x(y-x)(x²+y²+xy)+y²z(y-x)+z³(y-x)+yz²(y-x)]

P(x,y,z)=(y-z)(y-x)[-x(x²+y²+xy)+y²z+z³+yz²]

P(x,y,z)=(y-z)(y-x)[-x³-xy²-x²y+y²z+z³+yz²]

P(x,y,z)=(y-z)(y-x)[z³-x³-x²y+yz²-xy²+y²z]

P(x,y,z)=(y-z)(y-x)[z³-x³+y(z²-x²)+y²(z-x)]

P(x,y,z)=(y-z)(y-x)[(z-x)(z²+x²+zx)+y(z-x)(z+x)+y²(z-x)]

P(x,y,z)=(y-z)(y-x)(z-x)[z²+x²+zx+y(z+x)+y²]

P(x,y,z)=(y-z)(y-x)(z-x)[z²+x²+zx+yz+yx+y²]

P(x,y,z)=(y-z)(y-x)(z-x)(x²+y²+z²+zx+yz+yx)