The given polynomial is `f(x) = (3x^(4)-15x^(3)+13x^(2)+25x-30).`
Since ` sqrt(5/3) and -sqrt(5/3) ` are the zeros of f(x), it follows that each one of `(x-sqrt(5/3)) and (x+sqrt(5/3))` is factor of f(x).
` :. (x-sqrt5/sqrt3)(x+sqrt5/sqrt3)=(x^(2)-5/3) =((3x^(2)-5))/3 ` is a factor of f(x).
Consequently , `(3x^(2)-5)` is a factor of f(x).
On dividing f(x) by `(3x^(2)-5)`, we get
`:. f(x) = 3x^(4) - 15x^(3)+13x^(2) +25x-30`
` = (3x^(2)-5)(x^(2)-5x+6)`
` = (sqrt3 x+sqrt5)(sqrt3 x-sqrt5)(x-2)(x-3).`
` :. f(x) = 0 rArr (sqrt3 x+sqrt5)=0 or (sqrt3 x-sqrt5) = 0`
` or (x-2)= 0 or (x-3) = 0`
` rArr x =- sqrt(5/3) or x = sqrt(5/3) or x = 2 or x = 3 .`
Hence, all zeros of f(x) are ` sqrt(5/3), -sqrt(5/3),` 2 and 3.
