To list all possible rational zeros of the polynomial $f(x) = 7x^3 + 52x^2 + 22x + 7$, we use the Rational Root Theorem.

The Rational Root Theorem states that any possible rational zero of a polynomial is of the form:

$\frac{p}{q}$

Where:

• $p$ is a factor of the constant term (in this case, 7), and
• $q$ is a factor of the leading coefficient (in this case, 7).

Step 1: Factors of the constant term (7)

The factors of 7 are: $p = \pm 1, \pm 7$

Step 2: Factors of the leading coefficient (7)

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

Step 3: Form all possible values of $\frac{p}{q}$

We now list all possible combinations of $\frac{p}{q}$:

$\frac{p}{q} = \pm 1, \pm 7, \pm \frac{1}{7}, \pm \frac{7}{7}$

Simplifying:

$\pm 1, \pm 7, \pm \frac{1}{7}$

Conclusion

The possible rational zeros of the polynomial $f(x) = 7x^3 + 52x^2 + 22x + 7$ are:

$\boxed{\pm 1, \pm 7, \pm \frac{1}{7}}$

Would you like to check which of these are actual zeros, or need further details?

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Here are 5 follow-up questions to deepen your understanding:

1. How do you test whether a possible rational zero is an actual zero?
2. Can this method also help identify irrational or complex roots?
3. What is the next step after listing the possible rational zeros?
4. How would you graphically interpret the rational zeros of a cubic function?
5. What is the significance of the leading coefficient in finding rational zeros?

Tip: Always try substituting possible zeros into the original function to verify which are actual solutions.