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Part 1

Determine the different possibilities for the numbers of​positive, negative, and nonreal complex zeros for the following function.

​f(x)equals=negative 5 x cubed plus x squared minus 2 x plus 4−5x3+x2−2x+4

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Part 1

What is the possible number of positive real zeros of this​function?

To determine the possible number of positive real zeros, we use **Descartes' Rule of Signs**. This rule states that the number of positive real zeros of a polynomial function is either equal to the number of sign changes between consecutive terms of \( f(x) \), or less than that by an even integer.

### Step 1: Analyze \( f(x) \)

Given function:

\[
f(x) = -5x^3 + x^2 - 2x + 4
\]

Now, let's observe the signs of each term:

- The sign of \( -5x^3 \) is **negative**.
- The sign of \( x^2 \) is **positive**.
- The sign of \( -2x \) is **negative**.
- The sign of \( 4 \) is **positive**.

### Step 2: Count the Sign Changes

1. From \( -5x^3 \) (negative) to \( x^2 \) (positive): **1 sign change**.
2. From \( x^2 \) (positive) to \( -2x \) (negative): **1 sign change**.
3. From \( -2x \) (negative) to \( 4 \) (positive): **1 sign change**.

There are a total of **3 sign changes**.

### Step 3: Possible Number of Positive Real Zeros

According to Descartes' Rule of Signs, the number of positive real zeros is:
- The number of sign changes: **3**
- Or less than that by an even integer: \( 3 - 2 = 1 \)

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

Would you like me to continue solving for negative and nonreal complex zeros or have any other questions?

### Related Questions:

1. How do we determine the number of negative real zeros using Descartes' Rule of Signs?
2. Can you explain how to find the nonreal complex zeros of a polynomial function?
3. What are the steps to use Descartes' Rule of Signs on the transformed function \( f(-x) \)?
4. Are there other methods besides Descartes' Rule to determine the nature of zeros in polynomials?
5. How can we verify the number of real zeros using the graph of the function?

**Tip:** Remember that Descartes' Rule of Signs provides possible numbers of real zeros, but further verification is often needed using graphing or solving techniques.

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros for the following function: f(x) = -5x^3 + x^2 - 2x + 4. What is the possible number of positive real zeros of this function?

Discover how to apply Descartes' Rule of Signs to determine the possible number of positive real zeros for the cubic polynomial function f(x) = -5x^3 + x^2 - 2x + 4. This step-by-step guide walks through sign analysis and helps students in Grades 10-12 understand polynomial behavior.

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