We are given the polynomial function \( f(x) = x^3 - 2x^2 - 13x - 10 \) and we know that \( f(-2) = 0 \), which indicates that \( x = -2 \) is a zero of the function.
To find all zeros of \( f(x) \), we can perform polynomial long division or synthetic division to divide \( f(x) \) by \( x + 2 \) (since \( x = -2 \) is a root).
Perform synthetic division:
We set up synthetic division using \( -2 \):
\[ \begin{array}{r|rrrr} -2 & 1 & -2 & -13 & -10 \ & & -2 & 8 & 10 \ \hline & 1 & -4 & -5 & 0 \ \end{array} \]
The coefficients of the quotient are \( 1 \), \( -4 \), and \( -5 \). So we can write:
\[ f(x) = (x + 2)(x^2 - 4x - 5) \]
Now we need to factor the quadratic \( x^2 - 4x - 5 \) further. We can do this by finding two numbers that multiply to \(-5\) (the constant term) and add to \(-4\) (the coefficient of \( x \)):
These numbers are \(-5\) and \(1\).
Thus, we can factor the quadratic as:
\[ x^2 - 4x - 5 = (x - 5)(x + 1) \]
Now substituting this back, we have:
\[ f(x) = (x + 2)(x - 5)(x + 1) \]
Now we can find the zeros by setting each factor equal to zero:
- \( x + 2 = 0 \) → \( x = -2 \)
- \( x - 5 = 0 \) → \( x = 5 \)
- \( x + 1 = 0 \) → \( x = -1 \)
Thus, the zeros of \( f(x) \) are:
\[ \boxed{-2, 5, -1} \]
These zeros correspond to the choice:
x = -2, -1, 5