Mathematics
Factorise the following:
x4 + 9x2y2 + 81y4
Factorisation
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Answer
x4 + 9x2y2 + 81y4 = x4 + 18x2y2 - 9x2y2 + 81y4
= (x2)2 + (2 × x2 × 9y2) + (9y2)2 - 9x2y2
We know that,
(a + b)2 = a2 + b2 + 2ab
∴ (x2)2 + (2 × x2 × 9y2) + (9y2)2 - 9x2y2 = (x2 + 9y2)2 - 9x2y2
= (x2 + 9y2)2 - (3xy)2
We know that,
a2 - b2 = (a + b)(a - b)
∴ (x2 + 9y2)2 - (3xy)2 = (x2 + 9y2 + 3xy)(x2 + 9y2 - 3xy).
Hence, x4 + 9x2y2 + 81y4 = (x2 + 9y2 + 3xy)(x2 + 9y2 - 3xy).
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