Given the following polynomial function, use Descartes Rule of Signs to determine the possible number of positive and negative real zeros.
𝑓(𝑥)=𝑥^3−5𝑥^2+33𝑥−29
Number of positive real zeros

Number of negative real zeros

To apply **Descartes' Rule of Signs** and determine the number of positive and negative real zeros of the polynomial function \( f(x) = x^3 - 5x^2 + 33x - 29 \), we follow these steps:

### 1. Number of Positive Real Zeros
Descartes' Rule of Signs states that the number of positive real zeros of a polynomial function corresponds to the number of sign changes in the function \( f(x) \), or is less than that by an even integer.

Let's look at the signs of the coefficients of \( f(x) = x^3 - 5x^2 + 33x - 29 \):

- \( x^3 \) has a positive coefficient (+),
- \( -5x^2 \) has a negative coefficient (-),
- \( +33x \) has a positive coefficient (+),
- \( -29 \) has a negative coefficient (-).

Thus, the signs of the terms are: \( +, -, +, - \).

The number of sign changes is:
- \( + \rightarrow - \) (1st change),
- \( - \rightarrow + \) (2nd change),
- \( + \rightarrow - \) (3rd change).

There are **3 sign changes** in \( f(x) \), which means the number of positive real zeros could be:
- 3, or
- 1 (since it could be reduced by 2).

Thus, the **possible number of positive real zeros** is **3 or 1**.

### 2. Number of Negative Real Zeros
To determine the number of negative real zeros, we apply Descartes' Rule of Signs to \( f(-x) \). Let's substitute \( -x \) into the polynomial and simplify:

\[
f(-x) = (-x)^3 - 5(-x)^2 + 33(-x) - 29
\]
\[
f(-x) = -x^3 - 5x^2 - 33x - 29
\]

Now, check the signs of the terms in \( f(-x) = -x^3 - 5x^2 - 33x - 29 \):
- \( -x^3 \) has a negative coefficient (-),
- \( -5x^2 \) has a negative coefficient (-),
- \( -33x \) has a negative coefficient (-),
- \( -29 \) has a negative coefficient (-).

The signs are: \( -, -, -, - \).

There are **no sign changes** in \( f(-x) \), which means the number of negative real zeros is **0**.

### Final Answer:
- **Number of positive real zeros:** 3 or 1.
- **Number of negative real zeros:** 0.

Would you like a deeper explanation or have any further questions?

Here are 5 related questions for further practice:
1. How does Descartes' Rule of Signs apply to polynomials of degree 4?
2. What is the significance of a polynomial having complex roots?
3. Can a polynomial with no sign changes in \( f(x) \) have real zeros?
4. How do multiplicities of roots affect the number of real zeros in Descartes' rule?
5. How can you determine the exact number of real zeros using synthetic division or graphing?

**Tip**: Descartes' Rule of Signs only tells you the *possible* number of positive and negative real zeros. To find the exact number, you'll need to use additional methods like factoring, graphing, or using the Rational Root Theorem.

Given the following polynomial function, use Descartes' Rule of Signs to determine the possible number of positive and negative real zeros. f(x) = x^3 - 5x^2 + 33x - 29. Number of positive real zeros and number of negative real zeros.

Learn how to apply Descartes' Rule of Signs to determine the possible number of positive and negative real zeros of the polynomial f(x) = x^3 - 5x^2 + 33x - 29. This solution explores polynomial sign changes and evaluates possible zeros. Suitable for high school students in Grades 10-12.

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