Polymomial and Rational Functions
Using the conjugate zeros theorem to find all zeros of a polynomial

Find all other zeros of $P(x)=x^{3}-x^{2}-4 x-6$, given that $-1-i$ is a zero.
(If there is more than one zero, separate them with commas.)

Step 1 ：The given polynomial is \(P(x)=x^{3}-x^{2}-4 x-6\) and it is given that \(-1-i\) is a zero of the polynomial.

Step 2 ：According to the conjugate zeros theorem, if a polynomial has real coefficients and a complex number is a zero of the polynomial, then its conjugate is also a zero of the polynomial. Therefore, the conjugate of \(-1-i\), which is \(-1+i\), is also a zero of the polynomial.

Step 3 ：To find the remaining zeros, we can perform polynomial division to divide \(P(x)\) by the quadratic factor corresponding to the pair of complex conjugate zeros. The quotient will be a polynomial of lower degree, and we can find its zeros by setting it equal to zero and solving for \(x\).

Step 4 ：After performing the polynomial division, we find that the remaining zero is \(3\).

Step 5 ：\(\boxed{\text{The zeros of the polynomial } P(x)=x^{3}-x^{2}-4 x-6 \text{ are } -1-i, -1+i, \text{ and } 3}\)