- Factorise:

**(i) 8y**^{3}**– 1 **

Solution:

= (2y)^{3} – 1^{3} [∵ a^{3} – b^{3} = (a – b) (a^{2} + ab + b^{2})]

= (2y – 1) [(2y)^{2} + 2y .1 + 1^{2}]

= (2y – 1) (4y^{2} + 2y + 1)

**(ii) 27x**^{3}**– 8 **

Solution:

= (3x)^{3} – 2^{3 }[∵ a^{3 }– b^{3} = (a – b) (a^{2} + ab + b^{2})]

= (3x – 2) [(3x)^{2} + 3x .2 + 2^{2}]

= (3x – 2) (9x^{2} + 6x + 4)

**(iii) x**^{3 }**+ 8y**^{3}

Solution:

= x^{3} + (2y)^{3} [∵ a^{3} + b^{3} = (a + b) (a^{2} – ab + b^{2})]

= (x + 2y) [x^{2} – x.2y + (2y)^{2}]

= (x + 2y) (x^{2 }– 2xy + 4y)^{2}

**(iv) 1 – x**^{3}

Solution:

= 1^{3} – x^{3} [∵a^{3 }– b^{3} = (a – b) (a^{2} + ab + b^{2})]

= (1 – x) (1^{2} + 1.x + x^{2})

= (1 – x) (1 + x + x^{2})

**(v) a**^{3}**b**^{3}**+ c**^{3}

Solution:

= (ab)^{3} + c^{3 }[∵ a^{3} + b^{3} = (a + b) (a^{2} – ab + b^{2})]

= (ab + c) [(ab)^{2} – ab .c + c^{2}]

= (ab + c) (a^{2}b^{2} – abc + c^{2})

**(vi) a**^{3}**b – **^{𝐛}**/ _{𝟔𝟒} **

Solution:

= b (a^{3}– ^{1}/_{64})

= b [a^{3} – (^{1}/_{4} )^{3}]

= b (a – ^{1}/_{4}) [a^{2} + a. ^{1}/_{4} + ( ^{1}/_{4} )^{2}]

= b (a – 14) (a^{2} + ^{a}/_{4} + ^{1}/_{16} )

**(vii) ****𝐚**^{3}**/**_{𝟖}**+ 1 **

Solution:

= ( ^{a}/_{2} )^{3} + 1^{3} [∵ a^{3} + b^{3} = (a + b) (a^{2} – ab + b^{2})]

= ( ^{a}/_{2} + 1) [( ^{a}/_{2} )^{2} – ^{a}/_{2} .1 + 1^{2}]

= ( ^{a}/_{2} + 1) [a^{2}/_{4}− ^{a}/_{3}+ 1]

**(viii) 3a**^{6}**– ****𝐛**^{𝟔}**/**_{𝟗}

Solution:

= 3(a^{6}− b^{6}/27)

= 3[(a^{2})^{3}− (b^{2}/_{3})^{3}]

= 3[(a^{2}− b^{2}/_{3}) (a^{2})^{3}− a^{2}.b^{2}/_{3} (b^{2}/_{3})^{2}

= 3(a^{2}− b^{2}/_{3})( a^{4}+a^{2 }. b^{2}/_{3}+ b^{4}/_{3} )

**(ix) 2a**^{3}**+ **^{𝟏}**/ _{𝟒} **

Solution:

= 2 (a^{3} + ^{𝟏}/_{𝟒} )

= 2 (a^{3} + ^{𝟏}/_{2} )^{3}

= 2 (a + ^{𝟏}/_{2} ) [ a^{2} – a. ^{𝟏}/_{2} + (^{𝟏}/_{𝟒} )^{2}]

= 2 (a + ^{𝟏}/_{2} ) (a^{2} – ^{a}/_{2} + ^{𝟏}/_{𝟒} )

**(x) x**^{3}**– 512 **

Solution:

= x^{3} – 8^{3} [∵ a^{3} – b^{3} = (a – b) (a^{2} + ab + b^{2})]

= (x – 8) (x^{2} + 8x + 64)

**(xi) 32x**^{3}**– 500 **

= 4 (8x^{3} – 125)

= 4 [(2x)^{3} – 5^{3}]

= 4 (2x – 5) [(2x)^{2} + 2x .5 – 5^{2}]

**= 4 (2x – 5) (4x**^{2}**+ 10 x + 25) **

**(xii) x**^{7}**+ xy**^{6 }

= x (x^{6} + y^{6})

= x [(x^{2})^{3} + (y^{2})^{3}]

= x [(x^{2} + y^{2}) { (x^{2})^{2} – x^{2} y^{2} + (y^{2})^{2}}]

**= x (x**^{2}**+ y**^{2}**) (x**^{4}**– x**^{2}**y**^{2}**+ y**^{2}**)**

** **

**(xiii) 2a**^{4 }**– 128a **

= 2a (a^{3} – 64)

= 2a (a^{3} – 4^{3})

= 2a (a – 4) (a^{2} + 4a + 4^{2})

**= 2a (a – 4) (a**^{2}**+ 4a + 16)**

**Factorise**

**(i) (1 – a)**^{3}**+ (3a)**^{3}

Solution:

Using x^{3} + y^{3} = (x + y) (x^{2} – xy + y^{2})

(1 – a)^{3} + (3a)^{3}

= (1 – a + 3a) [(1 – a)^{2} – (1 – a) 3a + (3a)^{2}]

= (1 + 2a) [(1 – a)^{2} – 3a + 3a^{2} + 9a^{2}]

= (1 + 2a) (1 + a^{2} – 2a – 3a + 12a^{2})

= (1 + 2a) (13a^{2} – 5a + 1)

**(ii) 8x**^{3}**– 27y**^{3}

Solution:

= (2x)^{3} – ( 3y)^{3 }

= ( 2x – 3y) [(2x)^{2} + 2x .3y + (3y)^{2}]

= (2x – 3y) (4x^{2} + 6xy + 9y^{2})

**(iii) z**^{4}**x**^{3}**+ 8y**^{3}**z**^{4}

Solution:

= z^{4 }(x^{3} + 8y^{3})

= z^{4} [x^{3} + (2y)^{3}]

= z^{4} (x + 2y) [x^{2} – x.2y + (2y)^{2}]

= z^{4} (x + 2y) [x^{2} – 2xy + 4y^{2}]

**(iv) 3(x + y)**^{3}**+ **^{𝟏}**/ _{9} **

**(xy)**

^{3}Solution:

= (x + y)^{3} + ^{1}/_{27} (xy)^{3}

= (x + y)^{3} + ( ^{xy}/_{3} )^{3}

= (x + y + ^{xy}/_{3} ) (x + y)^{2} – [ (x + y) ( ^{xy}/_{3} )+ ( ^{xy}/_{3})^{2}]

= (x + y + ^{xy}/_{3} ) (x^{2} + y^{2 }+ 2xy – (x + y)( ^{xy}/_{3}) + x^{2}y^{2}/_{9} )

**(v) x**^{6}**+ y**^{6}

Solution:

= (x^{2})^{3} – (y^{2})^{3}

= (x^{2} – y^{2}) [(x^{2})^{3 }+ x^{2} y^{2}– (y^{2})^{3}]

= (x^{2} – y^{2}) (x^{4} + x^{2} y^{2} + y^{4})

= (x + y) (x – y) (x^{2} + x y + y^{2}) (x^{2} – xy + y^{2})

** **

**(vi) a**^{3}**– 2√****𝟐 ****b**^{3}

Solution:

= a^{3} – (√2 b)^{3 }

= (a –√2b) [a^{2} + a.√2 b + √2 b^{2}]

= (a –√𝟐 b) (a^{2} + √𝟐 ab + √𝟐b^{2})

**Factorize the following**

**(i) x**^{6}**– 26x**^{3}**– 27**

Solution:

put x^{3} = a

a^{3} – 26a – 27

= a^{3} – 27a + a – 27

= a ( a – 27) + 1 (a – 27)

= (a – 27) ( a + 1)

= (x^{3} – 27) (x^{3} + 1)

= (x – 3) (x^{2} + 3x + 9) (x – 1) (x^{2} – x + 1)

**(ii) z**^{6}**– 63z**^{3}**– 64 **

Solution:

put z^{3} = a

a^{3} – 63a – 64

= a^{3} – 64a + a – 64

= a (a – 64) + 1 (a – 64)

= (a – 64) (a + 1)

= (z^{3} – 64) (z^{3} + 1)

= (z – 4) (z^{2} + 4z + 16) (z + 1) (z^{2} – z + 1)

**(iii) a**^{3}**– b**^{3}**– a + b **

Solution:

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

= (a – b) [ a^{2} + ab + b^{2} – 1]

**(iv) x**^{6}**+ 7x**^{3}**– 8 **

Solution:

put x^{3} = a

a^{3} + 7a – 8

= a^{3} + 8a – a – 8

= a (a + 8) – 1 (a + 8)

= (a – 1) (a + 8)

= (x^{3} – 1) (x^{3} + 8)

= (x – 1) (x^{2 }+ x – 1) (x + 2) (x^{2} – 2x + 4)

**(v) a**^{3}**– **^{𝟏}**/ _{𝐚}**

^{𝟑}**– 2a +**

**𝟐𝐚**

Solution:

= (a – ^{𝟏}/_{𝐚}) (a^{2} + a.^{ 𝟏}/_{𝐚}+ ^{𝟏}/_{𝐚}^{2} ) – 2 (a – ^{𝟏}/_{𝐚} )

= (a – ^{𝟏}/_{𝐚} ) [a^{2 }+ 1 + ^{𝟏}/_{𝐚}^{2} – 2]

= (a – ^{𝟏}/_{𝐚} ) (a^{2} + ^{𝟏}/_{𝐚}^{𝟐} – 1 )

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