We will discuss here about the expansion of (a ± b)(a\(^{2}\) ∓ ab + b\(^{2}\)).
(a + b)(a\(^{2}\) - ab + b\(^{2}\)) = a(a\(^{2}\) - ab + b\(^{2}\)) + b(a\(^{2}\) - ab + b\(^{2}\))
= a\(^{3}\) - a\(^{2}\)b + ab\(^{2}\) + ba\(^{2}\) - ab\(^{2}\) + b\(^{3}\)
= a\(^{3}\) + b\(^{3}\).
(a - b)(a\(^{2}\) + ab + b\(^{2}\)) = a(a\(^{2}\) + ab + b\(^{2}\)) - b(a\(^{2}\) + ab + b\(^{2}\))
= a\(^{3}\) + a\(^{2}\)b + ab\(^{2}\) - ba\(^{2}\) - ab\(^{2}\) - b\(^{3}\)
= a\(^{3}\) - b\(^{3}\).
Problems on simplification of (a ± b)(a\(^{2}\) ∓ ab + b\(^{2}\))
1. Simplify: (2x + y)(4x\(^{2}\) – 2xy + y\(^{2}\))
Solution:
(2x + y)(4x\(^{2}\) – 2xy + y\(^{2}\))
= (2x + y){(2x)\(^{2}\) – (2x)y + y\(^{2}\)}
= (2x)\(^{3}\) + y\(^{3}\), [Since, (a + b)(a\(^{2}\) - ab + b\(^{2}\)) = a\(^{3}\) + b\(^{3}\)].
= 8x\(^{3}\) + y\(^{3}\).
2. Simplify: (x - \(\frac{1}{x}\))(x\(^{2}\) + 1 + \(\frac{1}{x^{2}}\)}
Solution:
(x - \(\frac{1}{x}\))(x\(^{2}\) + 1 + \(\frac{1}{x^{2}}\)}
= (x - \(\frac{1}{x}\)){x\(^{2}\) + x ∙ \(\frac{1}{x}\) + (\(\frac{1}{x}\))\(^{2}\)}
= x\(^{3}\) - \(\frac{1}{x^{3}}\), [Since, (a - b)(a\(^{2}\) + ab + b\(^{2}\)) = a\(^{3}\) - b\(^{3}\)].
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