Let's go through each question.

Question 9

To find a polynomial function with the given zeros, we need to construct a polynomial whose roots are: $-\frac{5}{3}, \sqrt{6}, -\sqrt{6}$

1. Construct the factors based on the roots:

   • The factor for $-\frac{5}{3}$ is $3x + 5$ (since we want integer coefficients).
   • The factors for $\sqrt{6}$ and $-\sqrt{6}$ are $(x - \sqrt{6})(x + \sqrt{6}) = x^2 - 6$.

2. Form the polynomial by multiplying these factors: $f(x) = (3x + 5)(x^2 - 6)$

3. Expand the expression: $f(x) = 3x \cdot x^2 - 3x \cdot 6 + 5 \cdot x^2 - 5 \cdot 6$ $f(x) = 3x^3 - 18x + 5x^2 - 30$ Rearranging terms gives: $f(x) = 3x^3 + 5x^2 - 18x - 30$

Checking the options, we can see that the answer matches: $f(x) = 3x^3 + 5x^2 - 18x - 30$

So, the correct choice is: $\text{Option 3: } f(x) = 3x^3 + 5x^2 - 18x - 30$

Question 10

The equation to solve here is: $x^2 - 1 = 0$

1. Rewrite as: $x^2 = 1$

2. Take the square root of both sides: $x = \pm 1$

So the roots are simply $x = 1$ and $x = -1$.

It looks like the provided answer choices may include complex solutions that aren't necessary here since the roots are real. Therefore, if there’s an option for real roots only, it should be chosen, or note that the roots are $1$ and $-1$.

Would you like more detailed steps for either question, or do you have any other specific questions?

Here are some related questions for further exploration:

1. How do we handle polynomial functions with irrational roots?
2. Why is it important for polynomial functions to have integer coefficients when specified?
3. What happens if a polynomial root is complex? How would the polynomial structure change?
4. How do we verify the degree of a polynomial with given roots?
5. What methods can be used to solve higher-degree polynomial equations?

Tip: When constructing polynomials from roots, always confirm that the roots match the polynomial's degree and that any required constraints (like integer coefficients) are satisfied.