Use Descartes' rule of signs to determine the possible number of positive and negative real zeros. $$ g\left(x\right)=9x^4-x^3-9x^2+8x+2 $$list the number of possible positive real zeros and negative real zeros 

Descartes' rule of signs helps us determine the possible number of positive and negative real zeros of a polynomial by examining the number of sign changes in the polynomial's terms.

### Positive Real Zeros
First, we examine the polynomial $ g(x) = 9x^4 - x^3 - 9x^2 + 8x + 2 $ to determine the possible number of positive real zeros. We look at the number of sign changes in the polynomial $ g(x) $.

The terms of $ g(x) $ are:
- $ 9x^4 $ (positive)
- $-x^3$ (negative)
- $-9x^2$ (negative)
- $8x$ (positive)
- $2$ (positive)

Let's count the sign changes:
- From $ 9x^4 $ to $-x^3$: 1 sign change (positive to negative)
- From $-x^3$ to $-9x^2$: 0 sign changes (negative to negative)
- From $-9x^2$ to $8x$: 1 sign change (negative to positive)
- From $8x$ to $2$: 0 sign changes (positive to positive)

There are 2 sign changes. According to Descartes' rule of signs, the number of positive real zeros is equal to the number of sign changes or less than it by an even number. Therefore, the possible number of positive real zeros is:
- 2
- 0 (since 2 - 2 = 0)

### Negative Real Zeros
To determine the possible number of negative real zeros, we substitute $ -x $ into $ g(x) $ and then count the sign changes in $ g(-x) $.

Let's substitute $ -x $ into $ g(x) $:
\[ g(-x) = 9(-x)^4 - (-x)^3 - 9(-x)^2 + 8(-x) + 2 \]
\[ g(-x) = 9x^4 + x^3 - 9x^2 - 8x + 2 \]

The terms of $ g(-x) $ are:
- $ 9x^4 $ (positive)
- $ x^3$ (positive)
- $-9x^2$ (negative)
- $-8x$ (negative)
- $2$ (positive)

Let's count the sign changes:
- From $ 9x^4 $ to $ x^3$: 0 sign changes (positive to positive)
- From $ x^3 $ to $-9x^2$: 1 sign change (positive to negative)
- From $-9x^2$ to $-8x$: 0 sign changes (negative to negative)
- From $-8x$ to $2$: 1 sign change (negative to positive)

There are 2 sign changes. According to Descartes' rule of signs, the number of negative real zeros is equal to the number of sign changes or less than it by an even number. Therefore, the possible number of negative real zeros is:
- 2
- 0 (since 2 - 2 = 0)

### Summary
- The possible number of positive real zeros is: 2 or 0.
- The possible number of negative real zeros is: 2 or 0.

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How do you find the exact number of real zeros for a given polynomial?
2. What are the steps to apply Descartes' rule of signs for polynomials?
3. Can you determine the multiplicity of real zeros using Descartes' rule of signs?
4. How does Descartes' rule of signs apply to complex roots?
5. What are the limitations of Descartes' rule of signs?
6. How do you determine the number of imaginary roots for a polynomial?
7. What is the significance of the Fundamental Theorem of Algebra in finding zeros?
8. How do you use synthetic division to verify potential zeros of a polynomial?

**Tip:** When analyzing polynomials, graphing the function can provide a visual understanding of the number and approximate location of real zeros.

Learn how to apply Descartes' Rule of Signs to determine the possible number of positive and negative real zeros for the polynomial g(x) = 9x^4 - x^3 - 9x^2 + 8x + 2. This problem involves analyzing sign changes in polynomial terms and understanding the concept of real zeros. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation