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Which of the following options is the correct factorization of the polynomial $3 a 3 b − 243 a b 3$ ?

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3 a b (a + 9 b) (a − 9 b)

3 a 2 b (a + 9 b) (a − 9 b)

3 a b 2 (a + 9 b) (a − 9 b)

3 a 2 b 2 (a + 9 b) (a − 9 b)

answer is A.

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Detailed Solution

The given polynomial is

3 a 3 b − 243 a b 3 .

The first step to solve these questions is to take common.
Thus,

3 a 3 b − 243 a b 3 = 3 a b (a 2 − 81 b 2) .

This can also be written as

3 a b [a 2 − (9 b) 2] .

We know that

a 2 − b 2 = a − b a + b .

Let’s consider

a = a and b = 9 b .

⇒ 3 a 3 b − 243 a b 3 = 3 a b (a + 9 b) (a − 9 b)

Thus, the factorized form of

3 a 3 b − 243 a b 3

will be

3 a b (a + 9 b) (a − 9 b) .

Therefore, option (1) is the correct option.

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