The question asks to determine the possible number of positive and negative real zeros of the polynomial function:

$f(x) = x^3 + 5x^2 - 48x - 252$

To determine the possible numbers of positive and negative real zeros, we use Descartes' Rule of Signs. This rule provides a method to estimate the number of positive and negative real roots of a polynomial based on the sign changes in the polynomial's terms.

1. Positive Real Zeros:

For positive real zeros, we analyze the sign changes in $f(x)$.

The function is: $f(x) = x^3 + 5x^2 - 48x - 252$

The signs of the coefficients are: $+, +, -, -$.

Count the number of sign changes:

• From $+x^3$ to $+5x^2$: no sign change.
• From $+5x^2$ to $-48x$: one sign change.
• From $-48x$ to $-252$: no sign change.

So, there is 1 sign change, meaning there is 1 possible positive real root.

2. Negative Real Zeros:

For negative real zeros, we substitute $-x$ into $f(x)$ and analyze the sign changes in $f(-x)$.

Substitute $-x$ into the polynomial: $f(-x) = (-x)^3 + 5(-x)^2 - 48(-x) - 252 = -x^3 + 5x^2 + 48x - 252$

This becomes: $f(-x) = -x^3 + 5x^2 + 48x - 252$

The signs of the coefficients are: $-, +, +, -$.

Count the number of sign changes:

• From $-x^3$ to $+5x^2$: one sign change.
• From $+5x^2$ to $+48x$: no sign change.
• From $+48x$ to $-252$: one sign change.

So, there are 2 sign changes, meaning there are either 2 or 0 negative real roots.

Conclusion:

According to Descartes' Rule of Signs:

• There is 1 possible positive real root.
• There are either 2 or 0 negative real roots.

Now, let's match this to the answer choices provided:

• Option B: "One positive and two negative roots" is the correct answer based on Descartes' Rule of Signs.

Do you want further details on this, or have any questions?

Here are 5 related questions to explore:

1. What is Descartes' Rule of Signs, and how does it work?
2. How can synthetic division be used to find exact roots after estimating them?
3. Can complex numbers be real roots in some cases, and how would that change the analysis?
4. How can the Rational Root Theorem assist in identifying specific roots?
5. What are the implications of a polynomial having no real roots?

Tip: Descartes' Rule of Signs gives only the possible number of real roots; the exact number requires further testing or factoring.