Prove that (a + b + c)3 - a3- b3 -c3= 3(a + b)(b + c)(c + a).
Answers
Answer:
ANSWER
L.H.S
=
(a−b)(a−c)
a
3
(b+c)
+
(b−c)(b−a)
b
3
(c+a)
+
(c−a)(c−b)
c
3
(a+b)
=
(a−b)(a−c)(b−c)
a
3
(b+c)(b−c)
+
(b−c)(b−a)(c−a)
b
3
(c+a)(c−a)
+
(c−a)(c−b)(a−b)
c
3
(a+b)(a−b)
(a−b)(b−c)(c−a)
−(a
3
b
2
−a
3
c
2
+b
3
c
2
−b
3
a
2
+c
3
a
2
−c
3
b
2
)
(a−b)(b−c)(c−a)
−(a
3
b
2
−b
3
a
2
+b
3
c
2
−a
3
c
2
+c
3
a
2
−c
3
b
2
)
(a−b)(b−c)(c−a)
−(a
2
b
2
(a−b)−c
3
(a
3
−b
3
)−c
3
(a
2
−b
2
))
(a−b)(b−c)(c−a)
(a−b)(c
2
(a
2
+b
2
+ab)−a
2
b
2
−c
3
(a+b))
=(a−b)
(a−b)(b−c)(c−a)
a
2
c
2
+c
2
b
2
+abc
2
−a
2
b
2
+ac
3
−bc
3
=(a−b)
(a−b)(b−c)(c−a)
a
2
c
2
−a
2
b
2
+b
2
c
2
−bc
3
+abc
2
−ac
3
=(a−b)
(a−b)(b−c)(c−a)
−a
2
(b
2
−c
2
)+bc
2
(b−c)+ac
2
(b−c)
=(a−b)(b−c)
(a−b)(b−c)(c−a)
−a
2
b−a
2
c+bc
2
+ac
2
=(a−b)(b−c)
(a−b)(b−c)(c−a)
−a
2
b+b
2
c−ac
2
+ac
2
=(a−b)(b−c)(c−a)
(a−b)(b−c)(c−a)
bc+ac+ab
=bc+ca+ab
R.H.S