Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. *h*(*x*) = 4*x*4 − 3*x* + 8

To apply **Descartes's Rule of Signs**, we analyze the changes in the signs of the coefficients in the given polynomial function to determine the possible number of positive and negative real zeros.

The function is:

\[
h(x) = 4x^4 - 3x + 8
\]

### Step 1: Positive real zeros

To find the possible number of positive real zeros, we look at the original function \( h(x) \). Now, we observe the changes in the signs of the coefficients of the terms:

- \( +4x^4 \) (positive)
- \( -3x \) (negative)
- \( +8 \) (positive)

The signs are: \( (+, -, +) \). We count the number of sign changes:
- From \( +4x^4 \) to \( -3x \): 1 change
- From \( -3x \) to \( +8 \): 1 change

There are **2 sign changes**, which means that there can be **2 or 0 positive real zeros** (since the number of positive real zeros is either the number of sign changes or less by an even number).

### Step 2: Negative real zeros

To find the possible number of negative real zeros, we substitute \( x \) with \( -x \) and simplify:

\[
h(-x) = 4(-x)^4 - 3(-x) + 8 = 4x^4 + 3x + 8
\]

The signs of the terms in \( h(-x) \) are:

- \( +4x^4 \) (positive)
- \( +3x \) (positive)
- \( +8 \) (positive)

The signs are: \( (+, +, +) \). There are **no sign changes**, which means there are **0 negative real zeros**.

### Summary of Possible Zeros:
- **Positive real zeros**: 2 or 0
- **Negative real zeros**: 0

Would you like further details or have any other questions?

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Here are 5 related questions to expand your understanding:
1. How can Descartes's Rule of Signs be used for higher degree polynomials?
2. Can Descartes's Rule of Signs determine the exact number of real zeros?
3. How does one verify whether the calculated possible zeros are actual zeros of the polynomial?
4. How does synthetic division help in finding real zeros once Descartes's Rule gives a clue?
5. How does the rule apply to complex zeros?

**Tip**: Descartes's Rule of Signs only provides possible numbers of real zeros, so using methods like the Rational Root Theorem or graphing can help in identifying exact zeros.

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. h(x) = 4x^4 − 3x + 8

Explore how to apply Descartes's Rule of Signs to the polynomial function h(x) = 4x^4 − 3x + 8. This guide provides a detailed analysis of sign changes and helps determine the possible number of positive and negative real zeros. Perfect for high school students studying polynomial functions.

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