To determine the correct equation for the polynomial $p(x)$, let's analyze the given information.

Problem Analysis

The polynomial $p(x)$ has zeros at:

1. $x = 0$
2. $x = -\frac{1}{6}$
3. $x = -3$

When a polynomial has a zero at a certain value $x = a$, it can be expressed with a factor of $(x - a)$. Therefore, the polynomial $p(x)$ should include the following factors:

1. $x$ (for the zero at $x = 0$)
2. $\left(x + \frac{1}{6}\right)$ (for the zero at $x = -\frac{1}{6}$)
3. $(x + 3)$ (for the zero at $x = -3$)

Factor Simplification

The factor $\left(x + \frac{1}{6}\right)$ can be rewritten as $\left(6x + 1\right)$ by multiplying by 6 to clear the fraction, leading to a polynomial form of: $p(x) = x (6x + 1)(x + 3)$

Answer Choice Evaluation

Looking at the choices, we see that:

• Option A: $p(x) = x(6x + 1)(x + 3)$

This matches our derived expression, so the correct answer is:

Answer: A

Would you like more details on the steps taken, or do you have any questions?

Related Questions

1. How do you determine the factors of a polynomial given its roots?
2. Why is multiplying by 6 necessary for clearing fractions in polynomial factors?
3. How can we confirm that a polynomial equation matches specific zeros?
4. What is the significance of polynomial roots in graphing the function?
5. How would the polynomial change if one of the zeros were at a different location?

Tip

When given polynomial roots with fractions, multiply factors to clear fractions and simplify the equation for easier comparison.