Given the function
P ( x )

The given function is:

$P(x) = x^3 + 2x^2 - 5x - 6$

Step 1: List all potential rational zeros (using the Rational Root Theorem)

The Rational Root Theorem states that potential rational zeros of a polynomial are given by:

$\frac{\text{factors of constant term}}{\text{factors of leading coefficient}}$

In this case:

• The constant term is $-6$, and its factors are: $\pm 1, \pm 2, \pm 3, \pm 6$.
• The leading coefficient is $1$, and its only factor is $\pm 1$.

Thus, the potential rational zeros are:

$\pm 1, \pm 2, \pm 3, \pm 6$

Potential rational zeros: $-6, -3, -2, -1, 1, 2, 3, 6$

Step 2: List the real zeros

To find the real zeros, we need to solve $P(x) = 0$ either by factoring or using numerical methods (such as synthetic division or the Rational Root Theorem). After testing the potential rational zeros:

• $P(1) = 1^3 + 2(1)^2 - 5(1) - 6 = 1 + 2 - 5 - 6 = -8 \quad \text{(not a zero)}$
• $P(-1) = (-1)^3 + 2(-1)^2 - 5(-1) - 6 = -1 + 2 + 5 - 6 = 0 \quad \text{(zero)}$

So, $x = -1$ is a real zero.

By factoring $P(x)$, we get:

$P(x) = (x + 1)(x - 2)(x + 3)$

Thus, the real zeros are $x = -1, 2, -3$.

Step 3: Write real zeros as ordered pairs

The real zeros of $P(x)$ as ordered pairs $(x, y)$ where $y = P(x) = 0$ are:

$(-1, 0), (2, 0), (-3, 0)$

---

Do you want more details or have any questions about the process?

Here are five questions to expand on this topic:

1. What is the Rational Root Theorem, and how is it used?
2. How do you perform synthetic division to test potential roots?
3. How would you find the complex roots of a cubic function?
4. Can you use the quadratic formula to find roots after factoring a cubic polynomial?
5. How do you graph the polynomial function and interpret its zeros?

Tip: When testing potential rational zeros, always start with smaller integers like $\pm 1$, as they are easiest to verify.