Simplification of (a + b + c)(a$^{2}$+b$^{2}$+c$^{2}$–ab–bc– ca)

We will discuss here about the expansion of (a + b + c)(a$^{2}$ + b$^{2}$ + c$^{2}$ – ab – bc – ca).

(a + b + c)(a$^{2}$ + b$^{2}$ + c$^{2}$ – ab – bc – ca)

= a(a$^{2}$ + b$^{2}$ + c$^{2}$ – ab – bc – ca) + b(a$^{2}$ + b$^{2}$ + c$^{2}$ – ab – bc – ca) + c(a$^{2}$ + b$^{2}$ + c$^{2}$ – ab – bc – ca)

= a$^{3}$ + ab$^{2}$ + ac$^{2}$ - a$^{2}$b – abc - ca$^{2}$ +ba$^{2}$ + b$^{3}$ + bc$^{2}$ - ab$^{2}$ – bc – bca + ca$^{2}$ + cb$^{2}$ + c$^{3}$ – cab - bc$^{2}$ - c$^{2}$a

= a$^{3}$ + b$^{3}$ + c$^{3}$ – 3abc.

Solved Example on Simplification of (a + b + c)(a$^{2}$ + b$^{2}$ + c$^{2}$  – ab – bc – ca)

1. Simplify: (x + 2y + 3z)(x$^{2}$ + 4y$^{2}$ + 9z$^{2}$ – 2xy – 6yz – 3zx)

Solution:

We know, (a + b + c)(a$^{2}$ + b$^{2}$ + c$^{2}$  – ab – bc – ca) = a$^{3}$ + b$^{3}$ + c$^{3}$ – 3abc.

Therefore, the given expression = (x + 2y + 3z){(x)$^{2}$ + (2y)$^{2}$ + (3z)$^{2}$ – (x)(2y) – (2y)(3z) – (3z)(x)}

= x$^{3}$ + (2y) $^{3}$ + (3z)$^{3}$ – 3 ∙ x ∙ 2y ∙ 3z

= x$^{3}$ + 8y$^{3}$ + 27z$^{3}$ – 18xyz.

Problem on simplification of (a + b + c)(a$^{2}$ + b$^{2}$ + c$^{2}$  – ab – bc – ca)

1. (x + y + 2z)(x$^{2}$ + y$^{2}$ + 4z$^{2}$ - xy - 2yz - 2zx)

2. (3a + 2b - c)(9a$^{2}$ + 4b$^{2}$ + c$^{2}$ - 6ab + 2b + 3ca)

Answer:

1. x$^{3}$ + y$^{3}$ + 8z$^{3}$ - 6xyz

2. 27a$^{3}$ + 8b$^{3}$ - c$^{3}$ + 18abc

9th Grade Math

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