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Factorise: $a^{12} - b^{12}$ .

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(a^{6} + b^{6}) (a + b) (a^{2} - ab - b^{2}) (a^{2} + ab + b^{2})

(a^{2} + b^{2}) (a^{4} + b^{4} - a^{2} b^{2}) (a + b) (a^{2} + b^{2} - ab) (a - b) (a^{2} + b^{2} + ab)

(a^{6} + b^{6}) (a - b) (a^{2} - ab + b^{2}) (a^{2} + ab + b^{2})

(a^{6} - b^{6}) (a + b) (a^{2} - ab + b^{2}) (a^{2} + ab + b^{2})

answer is B.

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Detailed Solution

a^{12} - b^{12} = {(a^{6})}^{2} - {(b^{6})}^{2}

= (a^{6} + b^{6}) (a^{6} - b^{6})

[As, (a^{2} - b^{2}) = (a + b) (a - b)]

= \{{(a^{2})}^{3} + {(b^{2})}^{3}\} \{{(a^{2})}^{3} - {(b^{2})}^{3}\}

= \{(a^{2} + b^{2}) ({(a^{2})}^{2} - a^{2} b^{2} + {(b^{2})}^{2})\} \{(a^{2} - b^{2}) ({(a^{2})}^{2} + a^{2} b^{2} + {(b^{2})}^{2})\}

[As, (a^{3} + b^{3}) = (a + b) (a^{2} - ab + b^{2}) And (a^{3} - b^{3}) = (a - b) (a^{2} + ab + b^{2})]

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

[As, (a^{2} - b^{2}) = (a + b) (a - b)]

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

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