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Factorise the polynomial $63 x 2 y 2 − 7$ completely and choose the option representing the product of the factors.

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7 (3 x y − 1) (3 x y + 1)

(21 x y − 7) (3 x y + 1)

(3 x y − 1) (21 x y + 7)

None of these

answer is A.

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Detailed Solution

We are given a polynomial expression as

63 x 2 y 2 − 7

.
Observing the expression, the HCF is identified as 7, because 63 is a multiple of 7.
So, factoring out the HCF from the expression, we have the new expression as follows,

63 x 2 y 2 − 7 = 7 9 x 2 y 2 − 1 = 7 3 x y 2 − 12

Now, it can be observed that the term

(3 x y) 2 − (1) 2

, is of the form

(a 2 − b 2)

, where

a = 3 x y

and

b = 1

.
So, using this identity, i.e.,

a 2 − b 2 = (a − b) (a + b)

, we have the completely factored form of the given polynomial expression as follows,

7 (3 x y − 1) (3 x y + 1)

So,

63 x 2 y 2 − 7

is completely factorised as

7 (3 x y − 1) (3 x y + 1)

.
Hence, the correct option is (1).

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