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One of the factors of $(a + b) 3 − 8$ is $(a + b − 2)$ . The other factor is:

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(a 2 + b 2 + 2 a b + 2 a + 4) .

(a 2 + b 2 + 2 a b + 2 a + b) .

(a 2 + b 2 + 4 a b + 2 a + 2 b + 4) .

(a 2 + b 2 + 2 a b + 2 a + 2 b + 4) .

answer is D.

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Detailed Solution

The given polynomial is given as

(a + b) 3 − 8

.
This polynomial can be written as,

(a + b) 3 − 8

⇒ (a + b) 3 − 23

.
Let us take

(a + b)

as p, say.
Then the polynomial reduces to the form

p 3 − 23

.
In the process of factorization, we generally tend to separate out the common terms from the polynomial to be factorized, and if it isn’t possible to take out any common terms, the polynomial is generally broken to form common terms.
Here,

p 3 − 23

(p − 2) (p 2 + 2 p + 22)

……. {

∵ a 3 − b 3 = (a − b) (a 2 + a b + b 2)

}

⇒ ((a + b) − 2) ((a + b) 2 + ((a + b) × 2) + 22) ⇒ (a + b − 2) ((a + b) 2 + 2 (a + b) + 4) ⇒ (a + b − 2) ((a 2 + b 2 + 2 a b) + 2 a + 2 b + 4) ⇒ (a + b − 2) (a 2 + b 2 + 2 a b + 2 a + 2 b + 4)

Therefore, the other factor is

(a 2 + b 2 + 2 a b + 2 a + 2 b + 4)

.
Hence, option 4 is correct.

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