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$^{3}$].

9th Grade Math

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