Q. 14.8( 5 Votes )

Let us write in t

Answer :

(i)


Expression 1: x + 9


Expression 2: x2 – 9x + 81


Product


= (x + 9)(x2 – 9x + 81)


= (x + 9)(x2 – 9x + 92)


= x3 + 93 [ (a + b)(a2 – ab + b2) = a3 + b3]


= x3 + 729


(ii)


Expression 1: 2a – 1


Product = 8a3 – 1


= (2a)3 - 13


= (2a – 1)[(2a)2 + (2a)(1) + (1)2]


[ a3 – b3 = (a – b)(a2 + ab + b2)]


= (2a – 1)(4a2 + 2a + 1)


Hence, Expression 2 is (4a2 + 2a + 1)


(iii)


Expression 1: 3 – 5c


Product = 27 – 125c3


= (3)3 – (5c)3


= (3 – 5c)(32 + 3(5c) + (5c)2) [ a3 – b3 = (a – b)(a2 + ab + b2)]


= (3 – 5c)(9 + 15c + 25c2)


Hence, Expression 2 is (9 + 15c + 25c2)


(iv)


Expression 1: (a + b + c)


Expression 2: (a + b)2 – (a + b)c + c2


Product


= (a + b + c)[(a + b)2 – (a + b)c + c2]


= [(a + b) + c][(a + b)2 – (a + b)c + c2]


[Using, a3 + b3 = (a + b)(a2 – ab + b2)]


If a = a + b, b = c


= (a + b)3 + c3


(v)


Expression 1: 3x


Expression 2: (2x – 1)2 – (2x – 1) (x + 1) + (x + 1)2


Product


= 3x [(2x – 1)2 – (2x – 1)(x + 1) + (x + 1)2]


As 3x can be written as 2x-1+x+1


= [2x-1+x+1] [(2x – 1)2 – (2x – 1) (x + 1) + (x + 1)2]


[Using, a3 + b3 = (a + b) (a2 – ab + b2)


Here a = 2x-1, b = x+1


= (2x – 1)3 + (x + 1)3


As (a-b)3 =a3 -3a2b+3ab2 – b3


(a + b)3=a3+3a2b+3ab2+b3


= (2x)3 – 1 – 3(2x)2(1) + 3(2x)(1)2 + x3 + 1 + 3x2 + 3x


= 8x3 – 1 – 12x2 + 6x + x3 + 1 + 3x2 + 3x


= 9x3 – 9x2 + 9x


(vi)


Expression 1:


Expression 2:


Product




[using, (a + b)3 = (a + b)(a2 – ab + b2)]




(vii)


Expression 1: 4a – 5b


Expression 2: 16a2 + 20ab + 25b2


Product


= (4a – 5b)(16a2 + 20ab + 25b2)


= (4a – 5b)((4a)2 + (4a)(5b) + (5b)2)


[Using, a3 – b3 = (a – b)(a2 + ab + b2)]


= (4a)3 – (5b)3


= 64a3 – 125b3


(viii)


Expression 2: a2b2 + abcd + c2d2


Product


= a3b3 – c3d3


= (ab)3 – (cd)3


= (ab – cd)[(ab)2 + (ab)(cd) + (cd)2]


[Using, a3 – b3 = (a – b)(a2 + ab + b2)]


= (ab – cd)(a2b2 + abcd + c2d2]


Hence, Expression 1 is (ab – cd)


(ix)


Expression 1: 1 – 4y


Product = 1 – 64y3


= 13 – (4y)3


= (1 – 4y)(12 + 1(4y) + (4y)2)


[ a3 – b3 = (a – b)(a2 + ab + b2)]


= (1 – 4y)(1 + 4y + 16y2)


Hence, Expression 2 is (1 + 4y + 16y2)


(x)


Expression 1: (2p + 1)


Product = 8(p – 3)3 + 343


= (2(p – 3))3 + 73


[Using, a3 – b3 = (a – b)(a2 + ab + b2)]


= [2(p – 3) + 7][(2(p – 3))2 + 2(p – 3)(7) + 72]


= (2p – 6 + 7)[4(p – 3)2 + 14p – 42 + 49]


= (2p + 1)(4(p2 – 6p + 9) + 14p + 7)


= (2p + 1)(4p2 – 24p + 36 + 14p + 7)


= (2p + 1)(4p2 – 10p + 43)


Hence, Expression 2 is (4p2 – 10p + 43)


(xi)


Expression 1 : m – p


Expression 2: (m + n)2 + (m + n)(n + p) + (n + p)2


Product


= (m – p)[(m + n)2 + (m + n)(n + p) + (n + p)2]


= [m + n – (n + p)]((m + n)2 + (m + n)(n + p) + (n + p)2]


[Using, a3 – b3 = (a – b)(a2 + ab + b2)]


= (m + n)3 – (n + p)3


= (m3 + n3 + 3m2n + 3mn2) – (n3 + p3 + 3n2p + 3np2)


= m3 + n3 + 3m2n + 3mn2 – n3 – p3 – 3n2p – 3np2


= m3 – p3 + 3m2n + 3mn2 – 3n2p – 3np2


(xii)


Expression 1: (3a - 2b)2 + (3a – 2b) × (2a – 3b) + (2a – 3b)2


Expression 2: (a + b)


Product


= (a + b)[(3a - 2b)2 + (3a – 2b) × (2a – 3b) + (2a – 3b)2]


= (3a – 2b – (2a – 3b))[(3a - 2b)2 + (3a – 2b) × (2a – 3b) + (2a – 3b)2]


= (3a – 2b)3 - (2a – 3b)3


[Now,


(a – b)3 = a3 – b3 – 3a2b + 3ab2]


= [(3a)3 – (2b)3 – 3(3a)2(2b) + 3(3a)(2b)2] – [(2a)3 – (3b)3 – 3(2a)2(3b) + 3(2a)(3b)2]


= [27a3 – 8b3 – 54a2b + 36ab2] – [8a3 – 27b3 – 36a2b + 54ab2]


= 27a3 – 8b3 – 54a2b + 36ab2 – 8a3 + 27b3 + 36a2b – 54ab2


= 19a3 + 19b3 – 18a2b – 18b2


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