Q. 24.8( 5 Votes )

Let us simplify using formula:

i.

ii.

iii.

iv.



v.



Answer :

i. (a + b)(a – b)(a2 + ab + b2)(a2 – ab + b2)


= (a + b)(a2 – ab + b2)(a – b)(a2 + ab + b2)


Using x3 – y3 = (x – y)(x2 + xy + y2)


x3 + y3 = (x + y)(x2 – xy + y2)


= (a3 + b3)(a3 – b3)


Now, using (a + b)(a – b) = a2 – b2


= a6 – b6


ii. (a – 2b)(a2 + 2ab + 4b2)(a3 + 8b3)


= (a – 2b)(a2 + a(2b) + (2b)2)(a3 + (2b)3)


Using x3 – y3 = (x – y)(x2 + xy + y2)


x3 + y3 = (x + y)(x2 – xy + y2)


= (a3 – (2b)3)(a3 + (2b)3)


Now, using (a + b)(a – b) = a2 – b2


= a6 – (2b)6


= a6 – 64b6


iii. (4a2 – 9)(4a2 – 6a + 9)(4a2 + 6a + 9)


= [(2a)2 – 32](4a2 – 6a + 9)(4a2 + 6a + 9)


Using x2 – y2 = (x – y)(x + y)


= (2a – 3)(2a + 3)(4a2 – 6a + 9)(4a2 + 6a + 9)


= (2a – 3)(4a2 + 6a + 9)(2a + 3)(4a2 – 6a + 9)


Using x3 – y3 = (x – y)(x2 + xy + y2)


x3 + y3 = (x + y)(x2 – xy + y2)


= [(2a)3 – 33][(2a)3 + 33]


= (2a)6 - 36


= 64a6 – 729


iv. (x – y)(x2 + xy + y2) + (y – z)(y2 + yz + z2) + (z – x)(z2 + zx + x2)


Using a3 – b3 = (a – b)(a2 + ab + b2)


= x3 – y3 + y3 – z3 + z3 – x3


= 0


v. (x + 1)(x2 - x + 1) + (2x – 1)(4x2 + 2x + 1) – (x – 1)(x2 + x + 1)


= (x + 1)(x2 - x + 1) + (2x – 1)((2x)2 + 2x + 1) – (x – 1)(x2 + x + 1)


Using x3 – y3 = (x – y)(x2 + xy + y2)


x3 + y3 = (x + y)(x2 – xy + y2)


= x3 + 1 + (2x)3 – 1 - (x3 – 1)


= x3 + 1 + 8x3 – 1 – x3 + 1


= 8x3 + 1


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