Question

Exercise 4.2
1. Prove that:
(x+y+z)3 = (y+z-x)2+(2+x-y)3+x+y-z)3+24xyz​

Answer 1

Answer:

What I tried is suppose a=(y+z−x), b=(z+x−y) and c=(x+y−z) and then noted that a+b+c=x+y+z. So the question statement reduced to (a+b+c)3−(a3+b3+c3). Then I tried to invoke the identity

(a3+b3+c3−3abc)=(a+b+c)(a2+b2+c2−ab−bc−ac)

by adding and subtracting 3abc in the question statement. After doing all this when I substituted back the values of a,b and c, I ended up with the initial question statement.

Any hints will be appreciated.