- Factorise:
(i) 8y3 – 1
Solution:
= (2y)3 – 13 [∵ a3 – b3 = (a – b) (a2 + ab + b2)]
= (2y – 1) [(2y)2 + 2y .1 + 12]
= (2y – 1) (4y2 + 2y + 1)
(ii) 27x3 – 8
Solution:
= (3x)3 – 23 [∵ a3 – b3 = (a – b) (a2 + ab + b2)]
= (3x – 2) [(3x)2 + 3x .2 + 22]
= (3x – 2) (9x2 + 6x + 4)
(iii) x3 + 8y3
Solution:
= x3 + (2y)3 [∵ a3 + b3 = (a + b) (a2 – ab + b2)]
= (x + 2y) [x2 – x.2y + (2y)2]
= (x + 2y) (x2 – 2xy + 4y)2
(iv) 1 – x3
Solution:
= 13 – x3 [∵a3 – b3 = (a – b) (a2 + ab + b2)]
= (1 – x) (12 + 1.x + x2)
= (1 – x) (1 + x + x2)
(v) a3 b3 + c3
Solution:
= (ab)3 + c3 [∵ a3 + b3 = (a + b) (a2 – ab + b2)]
= (ab + c) [(ab)2 – ab .c + c2]
= (ab + c) (a2b2 – abc + c2)
(vi) a3b – 𝐛/𝟔𝟒
Solution:
= b (a3– 1/64)
= b [a3 – (1/4 )3]
= b (a – 1/4) [a2 + a. 1/4 + ( 1/4 )2]
= b (a – 14) (a2 + a/4 + 1/16 )
(vii) 𝐚3/𝟖 + 1
Solution:
= ( a/2 )3 + 13 [∵ a3 + b3 = (a + b) (a2 – ab + b2)]
= ( a/2 + 1) [( a/2 )2 – a/2 .1 + 12]
= ( a/2 + 1) [a2/4− a/3+ 1]
(viii) 3a6 – 𝐛𝟔/𝟗
Solution:
= 3(a6− b6/27)
= 3[(a2)3− (b2/3)3]
= 3[(a2− b2/3) (a2)3− a2.b2/3 (b2/3)2
= 3(a2− b2/3)( a4+a2 . b2/3+ b4/3 )
(ix) 2a3 + 𝟏/𝟒
Solution:
= 2 (a3 + 𝟏/𝟒 )
= 2 (a3 + 𝟏/2 )3
= 2 (a + 𝟏/2 ) [ a2 – a. 𝟏/2 + (𝟏/𝟒 )2]
= 2 (a + 𝟏/2 ) (a2 – a/2 + 𝟏/𝟒 )
(x) x3 – 512
Solution:
= x3 – 83 [∵ a3 – b3 = (a – b) (a2 + ab + b2)]
= (x – 8) (x2 + 8x + 64)
(xi) 32x3 – 500
= 4 (8x3 – 125)
= 4 [(2x)3 – 53]
= 4 (2x – 5) [(2x)2 + 2x .5 – 52]
= 4 (2x – 5) (4x2 + 10 x + 25)
(xii) x7 + xy6
= x (x6 + y6)
= x [(x2)3 + (y2)3]
= x [(x2 + y2) { (x2)2 – x2 y2 + (y2)2}]
= x (x2 + y2) (x4 – x2 y2 + y2)
(xiii) 2a4 – 128a
= 2a (a3 – 64)
= 2a (a3 – 43)
= 2a (a – 4) (a2 + 4a + 42)
= 2a (a – 4) (a2 + 4a + 16)
- Factorise
(i) (1 – a)3 + (3a)3
Solution:
Using x3 + y3 = (x + y) (x2 – xy + y2)
(1 – a)3 + (3a)3
= (1 – a + 3a) [(1 – a)2 – (1 – a) 3a + (3a)2]
= (1 + 2a) [(1 – a)2 – 3a + 3a2 + 9a2]
= (1 + 2a) (1 + a2 – 2a – 3a + 12a2)
= (1 + 2a) (13a2 – 5a + 1)
(ii) 8x3 – 27y3
Solution:
= (2x)3 – ( 3y)3
= ( 2x – 3y) [(2x)2 + 2x .3y + (3y)2]
= (2x – 3y) (4x2 + 6xy + 9y2)
(iii) z4 x3 + 8y3 z4
Solution:
= z4 (x3 + 8y3)
= z4 [x3 + (2y)3]
= z4 (x + 2y) [x2 – x.2y + (2y)2]
= z4 (x + 2y) [x2 – 2xy + 4y2]
(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 ) (x2 + y2 + 2xy – (x + y)( xy/3) + x2y2/9 )
(v) x6 + y6
Solution:
= (x2)3 – (y2)3
= (x2 – y2) [(x2)3 + x2 y2– (y2)3]
= (x2 – y2) (x4 + x2 y2 + y4)
= (x + y) (x – y) (x2 + x y + y2) (x2 – xy + y2)
(vi) a3 – 2√𝟐 b3
Solution:
= a3 – (√2 b)3
= (a –√2b) [a2 + a.√2 b + √2 b2]
= (a –√𝟐 b) (a2 + √𝟐 ab + √𝟐b2)
- Factorize the following
(i) x6 – 26x3 – 27
Solution:
put x3 = a
a3 – 26a – 27
= a3 – 27a + a – 27
= a ( a – 27) + 1 (a – 27)
= (a – 27) ( a + 1)
= (x3 – 27) (x3 + 1)
= (x – 3) (x2 + 3x + 9) (x – 1) (x2 – x + 1)
(ii) z6 – 63z3 – 64
Solution:
put z3 = a
a3 – 63a – 64
= a3 – 64a + a – 64
= a (a – 64) + 1 (a – 64)
= (a – 64) (a + 1)
= (z3 – 64) (z3 + 1)
= (z – 4) (z2 + 4z + 16) (z + 1) (z2 – z + 1)
(iii) a3 – b3 – a + b
Solution:
= (a – b) (a2 + ab + b2) – (a – b)
= (a – b) [ a2 + ab + b2 – 1]
(iv) x6 + 7x3 – 8
Solution:
put x3 = a
a3 + 7a – 8
= a3 + 8a – a – 8
= a (a + 8) – 1 (a + 8)
= (a – 1) (a + 8)
= (x3 – 1) (x3 + 8)
= (x – 1) (x2 + x – 1) (x + 2) (x2 – 2x + 4)
(v) a3 – 𝟏/𝐚𝟑 – 2a + 𝟐𝐚
Solution:
= (a – 𝟏/𝐚) (a2 + a. 𝟏/𝐚+ 𝟏/𝐚2 ) – 2 (a – 𝟏/𝐚 )
= (a – 𝟏/𝐚 ) [a2 + 1 + 𝟏/𝐚2 – 2]
= (a – 𝟏/𝐚 ) (a2 + 𝟏/𝐚𝟐 – 1 )
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