Factors of: $54 a^{3} - 250 b^{3}$ are :

2

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3 a - 5 b

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5 a - 3 b

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9 a^{2} + 15 a b + 25 b^{2}

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Solution

The correct options are
A $2$
B $3 a - 5 b$
D $9 a^{2} + 15 a b + 25 b^{2}$
$54 a^{3} - 250 b^{3}$
$= 2 (27 a^{3} - 125 a^{3})$
$= 2 (3^{3} a^{3} - 5^{3} a^{3})$
$= 2 {{(3 a)}^{3} - {(5 b)}^{3}}$
Using $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ , we get
$54 a^{3} - 250 b^{3}$
$= 2 (3 a - 5 b) ((3 a)^{2} + 3 a \times 5 b + (5 b)^{2})$
$= 2 (3 a - 5 b) (9 a^{2} + 15 a b + 25 b^{2})$
Thus, the factors are $2, (3 a - 5 b)$ and $(9 a^{2} + 15 a b + 25 b^{2})$
Hence, A,B and D are correct.

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