Use the rational zeros theorem to list all possible rational zeros of $$ g\left(x\right)=-2x^4-5x^3+8x^2-4x+9 $$

The Rational Zeros Theorem states that any rational zero of a polynomial $ P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 $ must be of the form $ \frac{p}{q} $, where $ p $ is a factor of the constant term $ a_0 $ and $ q $ is a factor of the leading coefficient $ a_n $.

For the polynomial $ g(x) = -2x^4 - 5x^3 + 8x^2 - 4x + 9 $, the constant term $ a_0 $ is 9, and the leading coefficient $ a_n $ is -2.

First, we list the factors of the constant term 9:
\[
\pm 1, \pm 3, \pm 9
\]

Next, we list the factors of the leading coefficient -2:
\[
\pm 1, \pm 2
\]

According to the Rational Zeros Theorem, the possible rational zeros are given by the ratio $ \frac{p}{q} $, where $ p $ is any factor of 9 and $ q $ is any factor of -2.

This results in the following possible rational zeros:
\[
\pm 1, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2}, \pm 9, \pm \frac{9}{2}
\]

Therefore, the possible rational zeros of $ g(x) = -2x^4 - 5x^3 + 8x^2 - 4x + 9 $ are:
\[
\pm 1, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2}, \pm 9, \pm \frac{9}{2}
\]

Would you like further details or have any questions?

Here are 8 related questions that expand on this information:
1. How do you determine the actual rational zeros from the list of possible rational zeros?
2. Can the Rational Zeros Theorem be applied to non-polynomial functions?
3. How would you use synthetic division to test each possible rational zero?
4. What is the significance of the Rational Zeros Theorem in polynomial root finding?
5. How do you factorize a polynomial after finding one rational zero?
6. What other methods can be used to find zeros of polynomials?
7. Can the Rational Zeros Theorem be used for polynomials with irrational coefficients?
8. How does the degree of the polynomial affect the number of possible rational zeros?

**Tip:** When testing possible rational zeros, synthetic division can be a quick method to determine if a candidate is indeed a zero.

Explore how to apply the Rational Zeros Theorem to find all possible rational zeros of the polynomial -2x^4 - 5x^3 + 8x^2 - 4x + 9. Learn the step-by-step process with examples and understand the significance of this theorem in polynomial root finding. Suitable for high school students and anyone studying polynomial functions.

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