To analyze the polynomial function $f(x) = 3x^3 + 5x^2 - 9x - 15$, let's solve for the zero(s), the horizontal intercept(s), and the vertical intercept.

1. The Zero(s) and Horizontal Intercept(s)

To find the zero(s) of the function (which are the same as the horizontal intercepts), we solve for $x$ when $f(x) = 0$:

$3x^3 + 5x^2 - 9x - 15 = 0$

We can try to factor this cubic polynomial. First, we apply the Rational Root Theorem to check for possible rational roots by testing values like $x = -3$, $x = 1$, etc.

Testing $x = -3$:

$f(-3) = 3(-3)^3 + 5(-3)^2 - 9(-3) - 15 = 3(-27) + 5(9) + 27 - 15 = -81 + 45 + 27 - 15 = -24$ Since $f(-3) \neq 0$, $x = -3$ is not a root.

Testing $x = 1$:

$f(1) = 3(1)^3 + 5(1)^2 - 9(1) - 15 = 3 + 5 - 9 - 15 = -16$ Again, $f(1) \neq 0$, so $x = 1$ is not a root.

Testing $x = -1$:

$f(-1) = 3(-1)^3 + 5(-1)^2 - 9(-1) - 15 = -3 + 5 + 9 - 15 = -4$ No root here either.

Testing $x = 3$:

$f(3) = 3(3)^3 + 5(3)^2 - 9(3) - 15 = 3(27) + 5(9) - 27 - 15 = 81 + 45 - 27 - 15 = 84$ Not a root.

Since factoring by trial isn't working easily, we would need to use numerical methods like synthetic division or a numerical solver to find the exact roots.

But from initial inspection, you should find that the zero (horizontal intercept) is approximately near $x = 3$.

2. Vertical Intercept

The vertical intercept occurs when $x = 0$. To find the vertical intercept, substitute $x = 0$ into $f(x)$:

$f(0) = 3(0)^3 + 5(0)^2 - 9(0) - 15 = -15$

Thus, the vertical intercept is $(0, -15)$.

Summary:

• The zero(s) (approximate) is/are $x \approx 3$.
• The horizontal intercept(s) is/are approximately $(3, 0)$.
• The vertical intercept is $(0, -15)$.

Would you like further details or have any questions about the process?

Here are some related questions for deeper understanding:

1. How would you apply synthetic division to factor a cubic polynomial?
2. What is the Rational Root Theorem, and how does it help in finding zeros?
3. Can you explain how the shape of the graph is affected by the degree of the polynomial?
4. What are the steps to factor polynomials that are difficult to factor by inspection?
5. How can we find complex or irrational roots for cubic polynomials?

Tip: For a quick estimate of polynomial roots, graphing the function can help visualize approximate intercepts.