The image describes a problem related to finding the possible rational zeros of a polynomial using the Rational Root Theorem. The polynomial given is:

$P(x) = 14x^3 + 7x^2 - 21x - 34$

Rational Root Theorem:

If a polynomial has integer coefficients, the possible rational zeros $\frac{p}{q}$ are of the form:

• $p$ is a factor of the constant term (here, -34)
• $q$ is a factor of the leading coefficient (here, 14)

Step 1: Find factors of the constant term (-34)

The factors of -34 are: $p = \pm 1, \pm 2, \pm 17, \pm 34$

Step 2: Find factors of the leading coefficient (14)

The factors of 14 are: $q = \pm 1, \pm 2, \pm 7, \pm 14$

Step 3: Form the possible rational roots

The possible rational roots are of the form $\frac{p}{q}$, where $p$ is any factor of -34 and $q$ is any factor of 14. This gives the following possible rational roots:

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{7}, \pm \frac{1}{14}, \pm 2, \pm \frac{2}{7}, \pm \frac{2}{14}, \pm 17, \pm \frac{17}{2}, \pm \frac{17}{7}, \pm \frac{17}{14}, \pm 34, \pm \frac{34}{2}, \pm \frac{34}{7}, \pm \frac{34}{14}$

Thus, the full list of possible rational zeros is:

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{7}, \pm \frac{1}{14}, \pm 2, \pm \frac{2}{7}, \pm \frac{2}{14}, \pm 17, \pm \frac{17}{2}, \pm \frac{17}{7}, \pm \frac{17}{14}, \pm 34, \pm \frac{34}{2}, \pm \frac{34}{7}, \pm \frac{34}{14}$

Do you have any questions or want further details on how to proceed with testing which of these are actual roots?

Related Questions:

1. How do you test which rational roots are valid using synthetic division?
2. What is the significance of the Rational Root Theorem in factoring polynomials?
3. How would the solution change if the polynomial had no real rational roots?
4. Can a polynomial have irrational roots? If so, how are they determined?
5. How does the Rational Root Theorem apply when the polynomial has complex coefficients?

Tip:

Always test the possible rational roots using substitution or synthetic division to identify actual zeros from the list generated by the Rational Root Theorem.