Question

If a + b + c = 0, then prove the following

(a) (b + c) (b − c) + a(a + 2b) = 0

(b) a(a² − bc) + b(b² − ca) + c(c² − ab) = 0

(c) a(b² + c²) + b(c² + a²) + c(a² + b²) = −3abc

(d)

Answer 1

It is given that a + b + c = 0.

(a) We need to show that

Consider the left hand side.

Now, a + b + c = 0

⇒ a + b = −c

Hence,

(b) We need to show that

Consider the left hand side.

Using a + b + c = 0, we get

Hence,

(c) We need to show that

Consider the left hand side.

Using a + b + c = 0, we get

Hence,

(d) We need to show that

Consider the left hand side.

We know that

Using a + b + c = 0, we get

Hence,