Q. 84.3( 212 Votes )

# Factorize each of the following:

(i)

(ii)

(iii)

(iv)

(v)

Answer :

**(i)**

Using identity,

(a + b)^{3} = a^{3} + b^{3} + 3a^{2}b + 3ab^{2}

8a^{3} + b^{3} + 12a^{2}b + 6ab^{2}

= (2a)^{3} + b^{3} + 3 (2a) (2b) + 3 (2a) (b)^{2}

= (2a + b)^{3}

= (2a + b) (2a + b) (2a + b)

**(ii)**

Using identity,

(a + b)^{3} = a^{3} + b^{3} + 3a^{2}b + 3ab^{2}

8a^{3} – b^{3} – 12a^{2}b + 6ab^{2}

= (2a)^{3} – b^{3} – 3 (2a)^{2}b + 3 (2a) (b)^{2}

= (2a – b)^{3}

= (2a – b) (2a – b) (2a – b)

**(iii)** Using identity,

(a - b)^{3} = a^{3} - b^{3} - 3a^{2}b + 3ab^{2}

27 – 125a^{3} – 135a + 225a^{2}

= 3^{3} – (5a)^{3} – 3 (3)^{2}(5a) + 3 (3) (5a)^{2}

= (3 – 5a)^{3}

= (3 – 5a) (3 – 5a) (3 – 5a)

**(iv)**

Using identity,

(a - b)^{3} = a^{3} - b^{3} - 3a^{2}b + 3ab^{2}

64a^{3} – 27b^{3} – 144a^{2}b + 108ab^{2}

= (4a)^{3} – (3b)^{3} – 3 (4a)^{2} (3b) + 3 (4a) (3b)^{2}

= (4a – 3b)^{3}

= (4a – 3b) (4a – 3b) (4a – 3b)

**(v)**

Using identity,

(a + b)^{3} = a^{3} + b^{3} + 3a^{2}b + 3ab^{2}

27p^{3} - - p^{2} + p

= (3p)^{3} – ()^{3} – 3 (3p)^{2} () + 3 (3p) ()^{2}

= (3p - )^{3}

= (3p - ) (3p - ) (3p - )

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