The equation provided in the image represents a polynomial function of the form:

$f(x) = -4x^5 + 10x^4 - 15x^3 + 6x^2 - 10x + 4$

To find the possible rational zeros of this polynomial, we can use the Rational Root Theorem. The Rational Root Theorem states that any rational root of a polynomial, expressed as $\frac{p}{q}$, must have the following properties:

• $p$ is a factor of the constant term (in this case, $4$).
• $q$ is a factor of the leading coefficient (in this case, $-4$).

Step 1: Identify the factors of the constant term and the leading coefficient.

• The constant term is $4$, and its factors are $\pm 1, \pm 2, \pm 4$.
• The leading coefficient is $-4$, and its factors are $\pm 1, \pm 2, \pm 4$.

Step 2: List the possible rational roots.

The possible rational zeros are the ratios of the factors of the constant term to the factors of the leading coefficient. Thus, the possible rational roots are:

$\pm 1, \pm 2, \pm 4, \pm \frac{1}{2}, \pm \frac{1}{4}$

Conclusion:

The possible rational zeros of the given polynomial are:

$\pm 1, \pm 2, \pm 4, \pm \frac{1}{2}, \pm \frac{1}{4}$

Would you like help testing these possible roots, or do you have any other questions? Here's a tip to consider:

Tip:

To efficiently test these possible rational roots, you can use synthetic division to check if any of them are actual zeros of the function.