If `sqrt2` is a zero of `p(x)=6x^3 +sqrt2x^2- 10x -4sqrt2 ,` find the remaining zeros

Explanation

Let `f(x) = 6x^(3)+sqrt(2)x^(2)-10x - 4sqrt(2)` and given that, `sqrt(2)` is one of the zeroes of `f(x)` i.e., `(x-sqrt(2))` is one of the factor of given cubic polynomial.
Now, using division algorithm,
`{:(" "ul(6x^(2)+7sqrt(2)+4" ")),( {:(x-sqrt(2))) " "6x^(3)+sqrt(2)x^(2)-10x-4sqrt(2)),(" "ul(underset(-)(6x^(3))underset(+)-6sqrt(2)x^(2)" ")),(" "7sqrt(2)x^(2)-10x-4sqrt(2)),(" "ul(underset(-)(7)sqrt(2)x^(2)underset(+)(-)14x" ") ),(" "4x-4sqrt(2)),(" "ul(underset(-)(4x)underset(-)(4sqrt(2)" "))),(" "xx):}`
`:. 6x^(3) + sqrt(2)x^(2)-10x - 4sqrt(2) = (6x^(2)+7sqrt(2)x+4) xx (x-sqrt(2))+0`
`[ :'` dividend = divisor `xx` quotient +remainder]
`=(x-sqrt(2))(6x^(2)+4sqrt(2)x+3sqrt(2)x+4)`
`= (x-sqrt(2)){sqrt(2)x(3sqrt(2)x+4)+ 1(3sqrt(2)x+4)}`
`= (x-sqrt(2)) {(3sqrt(2)x+4) (sqrt(2)x+1)}`
`= (x-sqrt(2))(sqrt(2)x+1) (3sqrt(2)x+4)`
So, its other zeores are `-(1)/(sqrt(2))` and `-(4)/(3sqrt(2))`