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)

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

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

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