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COMEDK2015MathematicsBinomial Theorem

The expression 1 (3 x+1) [ ( 1+ 3 x+1 2 )⁷- ( 1- 3 x+1 2 )⁷ ] is a polynomial in x of degree

Options

  1. A7
  2. B5
  3. C4
  4. D3

Correct answer

D. 3

Step-by-step solution

We have, aligned & 1 (3 x+1) ( 1+ 3 x+1 2 )⁷- ( 1- 3 x+1 2 )⁷ &= 1 2⁷ (3 x+1) (1+ (3 x+1) )⁷-(1- (3 x+1) )⁷ ...(i) aligned Now, (1+ 3 x+1 )⁷-(1- 3 x+1 )⁷ aligned & [ 7 0 + 7 1 (3 x+1) + 7 2 ( 3 x+1 )²+ . &- [ 7 0 - 7 1 3 x+1 + 7 7 ( 3 x+1 )⁷ ] aligned aligned &=2 [ 7 1 ( 3 x+1 )+ 7 3 ( 3 x+1 )³- 7 7 ( 3 x+1 )⁷ ] & . + 7 5 ( 3 x+1 )⁵+ 7 7 ( 3 x+1 )⁷ ] &=2 3 x+1 [7+35(3 x+1) aligned .+21(3 x+1)²+(3 x+1)³ ] Now, putting above value in Eq. (i), so the given expression becomes 1 2⁶ [42+105 x+21(3 x+1)²+(3 x+1)³ ] So, de

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