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AP EAMCET2015MathematicsBinomial Theorem

If |x| < 1 , then the coefficient of x^5 in the expansion of 3 x (x-2)(x+1) is

Options

  1. A33 32
  2. B-33 32
  3. C31 32
  4. D-31 32

Correct answer

B. -33 32

Step-by-step solution

Given, 3 x (x-2)(x+1) can be written as, aligned & 3 x=A(x+1)+B(x-2) & At x=2, & A= 3(2) 2+1 A= 3(2) 3 =A=2 & At x=-1, B= 3(-1) -1-2 & B= -3 -3 B=1 aligned Substitute the values of A and B in Eq. (i), we get aligned & 3 x (x-2)(x+1) = 2 x-2 + 1 x+1 & = 2 -2 (1- x 2 ) + 1 (x+1) =- (1- x 2 )⁻¹+(1+x)⁻¹ & =- [1+ x 2 + ( x 2 )^2+ + ( x 2 )^5+ . ] aligned + (1-x+x^2-x^3+x^4-x^5+ . ) Coefficient of x^5=- ( 1 2 )^5-1= -1 32 -1=- 33 32

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