O Polynomial and Rational Functions
Using the rational zeros theorem to find all zeros of a polynomial: Irration...
? QUESTION
The function below has at least one rational zero.
Use this fact to find all zeros of the function.
\[
h(x)=2 x^{3}-3 x^{2}-12 x-5
\]

If there is more than one zero, separate them with commas. Write exact values, not decimal approximations.

Step 1 ：The Rational Root Theorem states that if a polynomial has a rational root, p/q (where p is a factor of the constant term and q is a factor of the leading coefficient), then p is a factor of the constant term and q is a factor of the leading coefficient.

Step 2 ：The constant term in the given polynomial is -5 and the leading coefficient is 2.

Step 3 ：The factors of -5 are ±1, ±5 and the factors of 2 are ±1, ±2.

Step 4 ：So, the possible rational roots of the polynomial are ±1, ±5, ±1/2, ±5/2.

Step 5 ：We can use synthetic division or direct substitution to test these possible roots.

Step 6 ：By direct substitution, we find that -1, -1/2, and 5 are the roots of the polynomial.

Step 7 ：\(\boxed{-1, -1/2, 5}\) are the zeros of the function \(h(x) = 2x^3 - 3x^2 - 12x - 5\).