Evaluate :

(i) (a+ 1)(a-1)(a^2 + 1)

(ii) (a + b)(a-b)(a^2 + b^2)

(iii) (2a-b)(2a +b)(4a^2 + b^2)

(iv) (3 - 2x)(3 + 2x)(9 + 4x^2)

(v) (3x - 4y)(3x + 4y)( 9x^2 + 16y^2)

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Solution

(i) $(a + 1) (a - 1) (a^{2} + 1) = [(a)^{2} - (1)^{2}] (a^{2} + 1)$

$= (a^{2} - 1) (a^{2} + 1)$

$= (a^{2} - 1) (a^{2} + 1)$

$= (a^{2})^{2} - (1)^{2}$

$= a^{4} - 1$ .

(ii) $(a + b) (a - b) (a^{2} + b^{2}) = (a^{2} - b^{2}) (a^{2} + b^{2})$

$= (a^{2})^{2} - (b^{2})^{2}$

$a^{4} - b^{4}$ .

(iii) $(2 a - b) (2 a + b) (4 a^{2} + b^{2})$

$= [(2 a)^{2} - (b)^{2}] (4 a^{2} + b^{2})$

$= (4 a^{2} - b^{2}) (4 a^{2} + b^{2})$

$= (4 a^{2})^{2} - (b^{2})^{2}$

$= 16 a^{4} - b^{4}$ .

(iv) $(3 - 2 x) (3 + 2 x) (9 + 4 x^{2}) = [(3)^{2} - (2 x)^{2}] (9 + 4 x^{2})$

$= (9 - 4 x^{2}) (9 + 4 x^{2})$

$= (9)^{2} - (4 x^{2})^{2}$

$= 81 - 16 x^{4}$ .

(v) $(3 x - 4 y) (3 x + 4 y) (9 x^{2} + 16 y^{2}) = [(3 x)^{2} - (4 y)^{2}] (9 x^{2} + 16 y^{2})$

$= (9 x^{2} - 16 y^{2}) (9 x^{2} + 16 y^{2})$

$= (9 x^{2})^{2} - (16 y^{2})^{2}$

$= 81 x^{4} - 256 y^{4}$ .

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