Find all the zeroes of the polynomial function f(x)=x3−5x2+6x−30
To find the zeroes of a polynomial function, we set the function equal to zero and solve for the values of x.
f(x) = x^3 - 5x^2 + 6x - 30
Setting f(x) equal to zero:
0 = x^3 - 5x^2 + 6x - 30
Now, we factor the expression if possible:
0 = (x - 3)(x^2 - 2x + 10)
The quadratic factor, x^2 - 2x + 10, does not factor further using real numbers. Therefore, the only real zero is x = 3.
Thus, the zeros of the polynomial function f(x) = x^3 - 5x^2 + 6x - 30 are x = 3.