How do you factor completely m^8 +m^4/16 + 1/256m8+m416+1256?
1 Answer
Sep 17, 2016
= (m^2-sqrt(3)/2m+1/4)(m^2+sqrt(3)/2m+1/4)(m^2-1/2m+1/4)(m^2+1/2m+1/4)=(m2−√32m+14)(m2+√32m+14)(m2−12m+14)(m2+12m+14)
Explanation:
Note that:
(a^2-kab+b^2)(a^2+kab+b^2) = a^4+(2-k^2)a^2b^2+b^4(a2−kab+b2)(a2+kab+b2)=a4+(2−k2)a2b2+b4
Let
Then:
m^8+m^4/16+1/256m8+m416+1256
= (m^2)^4+(m^2)^2(n^2)^2+(n^2)^4=(m2)4+(m2)2(n2)2+(n2)4
= ((m^2)^2-(m^2)(n^2)+(n^2)^2)((m^2)^2+(m^2)(n^2)+(n^2)^2)=((m2)2−(m2)(n2)+(n2)2)((m2)2+(m2)(n2)+(n2)2)
= (m^4-m^2n^2+n^4)(m^4+m^2n^2+n^4)=(m4−m2n2+n4)(m4+m2n2+n4)
= (m^2-sqrt(3)mn+n^2)(m^2+sqrt(3)mn+n^2)(m^2-mn+n^2)(m^2+mn+n^2)=(m2−√3mn+n2)(m2+√3mn+n2)(m2−mn+n2)(m2+mn+n2)
= (m^2-sqrt(3)/2m+1/4)(m^2+sqrt(3)/2m+1/4)(m^2-1/2m+1/4)(m^2+1/2m+1/4)=(m2−√32m+14)(m2+√32m+14)(m2−12m+14)(m2+12m+14)
