# Simplification of (a ± b)(a$$^{2}$$ ∓ ab + b$$^{2}$$)

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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