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  •  Find all zeroes of the polynomial  if two of its zeroe

 Find all zeroes of the polynomial

(2x^4-9 x ^3 + 5 x^2 + 3 x -1 ) if two of its zeroes are

(2+ \sqrt 3) \: \: and\: \: ( 2- \sqrt 3)

 

 

 
 
 
 
 

Answers (1)

Let P(x) = (2x^4-9 x ^3 + 5 x^2 + 3 x -1 )

It is given that (2+ \sqrt 3) \: \: and\: \: ( 2- \sqrt 3) are two zeros of p (x) 

 so, 

p(x) = \left [ x -(2+ \sqrt 3) \right ] \left [x - ( 2- \sqrt3) \right ]g (x) \\\\ p (x) = ( x ^2 - 4x + 1 )g(x) \\\\ g ( x) = p (x) / ( x ^2 - 4 x + 1 )\\\\


f (x) = (x^2-4x + 1 ) (2x^2 -x-1 )

other two zeros of  p(X) are the zeros of the polynomial 2x^2-x-1

we have ,

2x^2-x-1=0 \\\\ 2x^2 -2x +x-1=0\\\\ 2x(x-1) + 1(x-1)= 0 \\\\ (2x+1)(x-1)

so, 

f(x) = (x-2-\sqrt3 )(x-2+\sqrt3 ) (2x+1) (x-1)

hence, the other two zeros are -1/2 and 1 

 

Posted by

Safeer PP

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