MathematicsIf two zeros of a polynomial x 3 −3 x 2 +2   are 1+ 3   and 1− 3  , then find the third zero of the polynomial.

If two zeros of a polynomial x 3 3 x 2 +2   are 1+ 3   and 1 3  , then find the third zero of the polynomial.


  1. A
    1  
  2. B
    1  
  3. C
    2  
  4. D
    2   

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    Solution:

    Given polynomial x 3 3 x 2 +2   has two of its zeroes as 1+ 3   and 1 3  . We need to find the remaining zeroes.
    By factor theorem, we have that if a   is the zero of a polynomial, then (xa)   is the factor of that polynomial.
    Since 1+ 3   and 1 3   are the two zeros of the polynomial, therefore, the respective factors are x1 3   and x1+ 3 .  
    Calculating the product of these factors we have,
    x1 3 x1+ 3 = x 2 x+x 3 xx 3 +13 x1 3 x1+ 3 = x 2 xx+0+13 x1 3 x1+ 3 = x 2 2x2  
    Now dividing the given polynomial x 3 3 x 2 +2   by x 2 2x2   we have,

    Quotient obtained is x1  .
    Equating the quotient equal to 0  , we have,
    x1=0 x=1  
    Hence, the remaining zero of the polynomial is 1  .
    Therefore, option 1 is correct.
     
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