Descartes' Rule of Signs helps determine the number of positive and negative real zeros of a polynomial function by analyzing the sign changes in the function and its transformed version $h(-x)$.

Step 1: Positive Real Zeros

Descartes' Rule of Signs states that the number of positive real zeros of a polynomial is either equal to the number of sign changes in the terms of $h(x)$, or less than that by an even integer.

We start by analyzing the given function:

$h(x) = 8x^3 - 9x + 2$

Now, examine the signs of the coefficients:

• $8x^3$ (positive)
• $-9x$ (negative)
• $+2$ (positive)

The sign changes are:

• From $8x^3$ (positive) to $-9x$ (negative): 1 change
• From $-9x$ (negative) to $+2$ (positive): 1 change

Thus, there are 2 sign changes in $h(x)$.

According to Descartes' Rule of Signs, the possible number of positive real zeros is either:

• 2 (the number of sign changes), or
• 0 (2 minus an even integer).

Step 2: Negative Real Zeros

To determine the number of negative real zeros, we apply Descartes' Rule of Signs to $h(-x)$. First, substitute $-x$ into the function:

$h(-x) = 8(-x)^3 - 9(-x) + 2 = -8x^3 + 9x + 2$

Now, analyze the signs of the coefficients in $h(-x)$:

• $-8x^3$ (negative)
• $+9x$ (positive)
• $+2$ (positive)

The sign changes are:

• From $-8x^3$ (negative) to $+9x$ (positive): 1 change
• From $+9x$ (positive) to $+2$ (positive): 0 changes

Thus, there is 1 sign change in $h(-x)$.

According to Descartes' Rule of Signs, the possible number of negative real zeros is either:

• 1 (the number of sign changes).

Conclusion

• The possible number of positive real zeros is either 2 or 0.
• The possible number of negative real zeros is 1.

Would you like more details or have any further questions? Here are some related questions for further exploration:

1. How can we verify the number of real zeros using a graph of the function?
2. What happens if there are complex roots in the function?
3. How would we apply the Rational Root Theorem to this function?
4. What if we wanted to approximate the real roots using numerical methods?
5. Can Descartes' Rule of Signs be used to determine the exact number of real zeros?

Tip: After using Descartes' Rule of Signs, you can further refine your understanding of the roots using the Intermediate Value Theorem or synthetic division.