Problem
ALG-B2-M06-P011 Fourth powers through p, q, r
#11
★★★★☆ Level 4 of 5
Prove the formula \[a^4+b^4+c^4=p^4-4p^2q+2q^2+4pr.\]
Hint. Use \(a^4+b^4+c^4=(a^2+b^2+c^2)^2-2(a^2b^2+b^2c^2+c^2a^2)\).
We have \(\sum a^2=p^2-2q\). Also \(a^2b^2+b^2c^2+c^2a^2=q^2-2pr\). Thus \[\sum a^4=(p^2-2q)^2-2(q^2-2pr)=p^4-4p^2q+2q^2+4pr.\]
A technical formula for degree \(4\).