ind all possible rational zeros for the polynomial function.

To find all possible **rational zeros** for a polynomial function, we apply the **Rational Root Theorem**. This theorem states that any rational solution (zero) of a polynomial equation:

\[
P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 = 0
\]

can be expressed in the form:

\[
\frac{p}{q}
\]

Where:
- **p** is a factor of the constant term \( a_0 \).
- **q** is a factor of the leading coefficient \( a_n \).

### Steps to find the possible rational zeros:

1. **Identify** the constant term \( a_0 \) and the leading coefficient \( a_n \).
2. **List** all the factors of \( a_0 \) (for \( p \)).
3. **List** all the factors of \( a_n \) (for \( q \)).
4. **Write all possible fractions** \( \frac{p}{q} \), where \( p \) is a factor of \( a_0 \) and \( q \) is a factor of \( a_n \), considering both positive and negative values.
5. **Test** these possible values by plugging them into the polynomial to check if they result in zero.

If you provide the specific polynomial, I can help you determine the possible rational zeros in detail.

Would you like to proceed with an example or share your polynomial?

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**Relative Questions:**
1. How is the Rational Root Theorem applied to quadratic polynomials?
2. What are the differences between rational and irrational zeros in polynomials?
3. How can synthetic division help in verifying rational roots?
4. How can you identify whether a polynomial has no rational zeros?
5. What happens if all possible rational zeros fail to solve the polynomial?

**Tip:** Always test possible rational zeros by substituting them back into the polynomial; if the polynomial equals zero, you have found a valid root!

Find all possible rational zeros for the polynomial function.

Learn how to find all possible rational zeros of a polynomial function by applying the Rational Root Theorem. This guide walks you through the steps of identifying the constant term and leading coefficient, listing their factors, and testing possible rational solutions. Ideal for students in Grades 10-12.

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