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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