Find all zeroes of the polynomial (2x4 - 9x3 + 5x2 + 3x-1) if two of its zeroes are (2 + 3) and (2 -3).


It is given that (2 + 3) and (2 -3) are two zeros of 2x4 - 9x3 + 5x2 + 3x-1

x - (2 + 3)x - (2-3 = (x-2-3)(x-2 +3)= (x-2)2 - (3)2 = x2 - 4x + 1 (x2- 4x + 1) is factor of f(x)

Let us now divide f(x) by x2-4x + 1

therefore, we have,

 f(x) = (x2- 4x + 1)(2x2-x-1)

Hence, other two zeros of f(x) are the zeros of the polynomial 2x2-x-1

We have, 

2x2-x-1 = 2x2- 2x+ x-1

= 2x (x-1) + 1(x-1)

= (2x+ 1)(x-1)

f(x) = (x-2-3) (x-2 +3) (2x+1)(2x-1)

Hence, the other two zeros are - 1/2 and 1.