To find the intercepts of the polynomial function $f(x) = x^3 + 5x^2 - 9x - 45$, we will compute both the x-intercepts and the y-intercept.

1. Finding the y-intercept:

The y-intercept occurs where $x = 0$.

$f(0) = (0)^3 + 5(0)^2 - 9(0) - 45 = -45$

Thus, the y-intercept is at $(0, -45)$.

2. Finding the x-intercepts:

The x-intercepts occur where $f(x) = 0$, so we need to solve the equation:

$x^3 + 5x^2 - 9x - 45 = 0$

We can try factoring the cubic polynomial using the Rational Root Theorem, which suggests possible rational roots by considering the factors of the constant term (-45) and the leading coefficient (1).

The possible rational roots are $\pm 1, \pm 3, \pm 5, \pm 9, \pm 15, \pm 45$.

Let's test $x = -5$:

$f(-5) = (-5)^3 + 5(-5)^2 - 9(-5) - 45 = -125 + 125 + 45 - 45 = 0$

Since $x = -5$ is a root, we can factor the polynomial as $(x + 5)$ and perform polynomial division to simplify.

Now divide $x^3 + 5x^2 - 9x - 45$ by $(x + 5)$ using synthetic or long division, which gives:

$x^3 + 5x^2 - 9x - 45 = (x + 5)(x^2 - 9)$

Next, factor the quadratic part:

$x^2 - 9 = (x - 3)(x + 3)$

Thus, the factored form of the polynomial is:

$f(x) = (x + 5)(x - 3)(x + 3)$

Setting each factor equal to 0:

$x + 5 = 0 \quad \Rightarrow \quad x = -5$ $x - 3 = 0 \quad \Rightarrow \quad x = 3$ $x + 3 = 0 \quad \Rightarrow \quad x = -3$

Therefore, the x-intercepts are $(-5, 0), (3, 0), (-3, 0)$.

Summary of Intercepts:

• y-intercept: $(0, -45)$
• x-intercepts: $(-5, 0), (3, 0), (-3, 0)$

Would you like further details on any of these steps or explanations?

Here are 5 related questions to expand on this problem:

1. How do we apply synthetic division to verify the factorization?
2. Can we graph the function to visually confirm the intercepts?
3. What is the significance of the Rational Root Theorem in solving cubic equations?
4. How does factoring help in solving higher-degree polynomials?
5. What if the polynomial had complex roots? How would we approach it?

Tip: The Rational Root Theorem is useful in testing possible rational roots and simplifies the factorization of polynomials.