If two zeroes of the polynomial f (x) = x^3-5x^2-2x+24 are root 3 and - root3 then find its third zero

Product of zeroes of a cubic polynomial is - d/a of cubic polynomial ax^3 + bx^2 + cx + d

So we know product of two zeroes therefore by putting value of product of two zeroes and taking third as a variable we get third zero as - 2

Therefore - 2 satisfies the equation and x+2 will be a factor of the equuation

By diving the above equation by x+2 we will get a quadratic rquation

I. e. x^2 - 7x + 12

Now by splitting the middle term you can find the other two zeroes

i. e. x^2 - 4x - 3x + 12

X (x - 4) - 3 (x - 4)

Therefore (x-3) (x-4) = x^2 - 4x - 3x + 12

Therefore the other two zeroes are 3 and 4

The zetoes for the cubic equation are - 2, 3, 4

To verify you can put these values in the equation and find answer = 0