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To determine the possible number of positive and negative real zeros of the polynomial \( g(x) = -6x^7 - x^4 - 7x^3 - 2x \) using Descartes' Rule of Signs, let's proceed step by step.

### Step 1: Determine Possible 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 \( g(x) \) or less than that by an even number.

#### Polynomial: 
\[
g(x) = -6x^7 - x^4 - 7x^3 - 2x
\]

We examine the signs of the coefficients in order:

- The coefficient of \( x^7 \) is \(-6\) (negative).
- The coefficient of \( x^4 \) is \(-1\) (negative).
- The coefficient of \( x^3 \) is \(-7\) (negative).
- The coefficient of \( x \) is \(-2\) (negative).

There are no sign changes in the sequence of terms. Therefore, **no positive real zeros** are possible.

### Step 2: Determine Possible Negative Real Zeros
To apply Descartes' Rule of Signs to negative real zeros, we substitute \( -x \) into the polynomial, i.e., we evaluate \( g(-x) \).

\[
g(-x) = -6(-x)^7 - (-x)^4 - 7(-x)^3 - 2(-x)
\]
Simplifying:
\[
g(-x) = 6x^7 - x^4 + 7x^3 + 2x
\]

Now, we examine the signs of the coefficients in \( g(-x) \):

- The coefficient of \( x^7 \) is \( 6 \) (positive).
- The coefficient of \( x^4 \) is \( -1 \) (negative).
- The coefficient of \( x^3 \) is \( 7 \) (positive).
- The coefficient of \( x \) is \( 2 \) (positive).

We observe two sign changes:
1. From \( 6x^7 \) (positive) to \( -x^4 \) (negative).
2. From \( -x^4 \) (negative) to \( 7x^3 \) (positive).

Thus, by Descartes' Rule of Signs, the number of negative real zeros could be 2 or 0 (2 minus an even number).

### Conclusion:
- Positive real zeros: 0 (since there are no sign changes in \( g(x) \)).
- Negative real zeros: 2 or 0 (due to 2 sign changes in \( g(-x) \)).

Would you like further clarification on any of these steps?

Here are 5 related questions for practice:
1. What is the general form of Descartes' Rule of Signs for determining the number of real zeros?
2. How would the number of real zeros change if there were three sign changes?
3. What happens if a polynomial has complex roots in addition to real ones?
4. Can a polynomial have more real roots than its degree?
5. What role does the degree of a polynomial play in determining the number of possible real roots?

**Tip**: Always check both \( g(x) \) and \( g(-x) \) for sign changes when applying Descartes' Rule of Signs to capture both positive and negative real roots.

Use Descartes' Rule of Signs to determine the possible number of positive and negative real zeros of the polynomial g(x) = -6x^7 - x^4 - 7x^3 - 2x.

Learn how to apply Descartes' Rule of Signs to determine the possible number of positive and negative real zeros of the polynomial g(x) = -6x^7 - x^4 - 7x^3 - 2x. This example breaks down the step-by-step process and offers practice questions for students in Grades 10-12.

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