To find the zeros of the function \( f(z) = 16z^3 - 80z^2 - z + 5 \), we need to solve the equation:

\[
16z^3 - 80z^2 - z + 5 = 0
\]

### Step 1: Use the Rational Root Theorem
The Rational Root Theorem suggests that any possible rational root \( \frac{p}{q} \) of the polynomial equation must have \( p \) as a factor of the constant term (5) and \( q \) as a factor of the leading coefficient (16).

- Factors of 5: \( \pm 1, \pm 5 \)
- Factors of 16: \( \pm 1, \pm 2, \pm 4, \pm 8, \pm 16 \)

Possible rational roots: \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{8}, \pm \frac{1}{16}, \pm 5, \pm \frac{5}{2}, \pm \frac{5}{4}, \pm \frac{5}{8}, \pm \frac{5}{16} \)

### Step 2: Test Possible Roots
We substitute each possible root into the polynomial to see if it equals zero.

After testing these values, you can find one real root, and then proceed to factor the cubic equation down to a quadratic, which can then be solved using the quadratic formula.

I'll calculate the roots to determine the zeros.

The zeros of the function \( f(z) = 16z^3 - 80z^2 - z + 5 \) are:

\[
z = -\frac{1}{4}, \quad z = \frac{1}{4}, \quad z = 5
\]

Would you like more details on the steps or have any questions?

Here are some related questions for further exploration:

1. How can the Rational Root Theorem help in identifying possible zeros for polynomials of higher degrees?
2. What is the significance of factoring cubic polynomials when finding zeros?
3. Can all cubic polynomials be factored into rational roots?
4. How does the discriminant of a quadratic equation affect the nature of its roots?
5. What are the implications of having repeated roots in a polynomial?

**Tip:** When solving higher-degree polynomials, finding one root first can simplify the process by allowing you to reduce the polynomial's degree.

Explore how to find zeros of the polynomial function 16z^3 - 80z^2 - z + 5 step-by-step. This problem involves using the Rational Root Theorem, factoring polynomials, and solving equations. Suitable for high school students studying polynomial equations.

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