How do you factor u^3 +v^3 +w^3 −3uvw = (u+v+w)((u+v+w)^2 −3(uv+vw+wu))?

u^3 +v^3 +w^3 −3uvw = (u+v+w)( (u+v+w)^2 −3(uv+vw+wu))

1 Answer
Shwetank Mauria
Jan 1, 2018

Please see below.

Explanation:

u^3+v^3+w^3-3uvw

= ul(u^3+v^3color(blue)(+3u^2v+3uv^2))+w^3-ul(3uvwcolor(red)(-3u^2v-3uv^2))

= ul((u+v)^3+w^3)-3uv(u+v+w)

= (u+v+w)((u+v)^2+w^2-(u+v)w)-3uv(u+v+w)

Here we have factorized (u+v)^3+w^3 using x^3+y^3=(x+y)(x^2+y^2-xy), where x=(u+v) and y=w. Observe that now we can take u+v+w ascommon and we get

(u+v+w)(u^2+2uv+v^2+w^2-uv-uw-3uv)

= (u+v+w)(u^2+v^2+w^2-uv-vw-uw)

These are standard factors of u^3+v^3+w^3-3uvw. Now also observe that

(u+v+w)^2=u^2+v^2+w^2+2uv+2vw+2uw

hence u^2+v^2+w^2-uv-vw-uw=(u+v+w)^2-3uv-3vw-3uw

= (u+v+w)^2-3(uv+vw+uw)

and hence u^3+v^3+w^3-3uvw

= (u+v+w)((u+v+w)^2-3(uv+vw+uw))

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