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Find the first four terms in the expansion of (1+x)^4(1-2x)^5 in ascending power of x
Math
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Find the first four terms in the expansion of (1+x)4(1-2x)5 in ascending power of x

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Joshua43
Asked by Joshua43rep:65
Nigeria
11 September 2020

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judziy
Judziy answeredrep:1.4K
University of Benin Nigeria
14 September 2020
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From binomial expansion and pascal triangle,

Thus, solving first for (1 + x)4

n = 4

(1 + x)4 = (40)(14x0) + (31)(13x1(22)(12x2(13)(11x3(04)(10x4)

= (1)(1) + (4)(x) + (6)(x2) + (4)(x3) + (1)(x4)

= 1 + 4x + 6x2 + 4x3 + x4

Similarly, (1 - 2x)5(50)(15(-2x)0) + (41)(14(-2x)1) (32)(13(-2x)2) + (23)(12(-2x)3) + (14)(11(-2x)4) + (05)(10(-2x)5)

= (1)(1) + (5)(-2x) + (10)(4x2) + (10)(-8x3) + (5)(16x4) + (1)(-32x5)

= 1 - 10x + 40x2 - 80x3 + 80x4 - 32x5

Thus, (1 + x)4(1 - 2x)5 = (1 + 4x + 6x2 + 4x3 + x4)(1 - 10x + 40x2 - 80x3 + 80x4 - 32x5)

1 - 10x + 40x2 - 80x3 + 80x4 - 32x5 + 4x - 40x2 + 160x3 - 320x4 + 320x5 - 128x+ 6x2 - 60x3 + 240x4 - 480x5 + 480x6 - 192x+ 4x3 - 40x4 + 160x5 - 320x6 + 320x7 - 128x+ x4 - 10x5 + 40x6 - 80x7 + 80x8 - 32x9

collect like terms,

= 1 - 10x + 4x + 40x2 - 40x+ 6x2- 80x3 + 160x- 60x+ 4x+ 80x- 320x+ 240x- 40x+ x- 32x+ 320x- 480x+ 160x- 10x- 128x+ 480x- 320x+ 40x- 192x+ 320x- 80x- 128x+ 80x- 32x9

= 1 - 6x + 6x2 + 24x3 - 39x- 42x5 + 72x6 + 48x7 + 208x8 - 32x9

Hence, teh first four terms are:

1 - 6x + 6x2 + 24x3

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