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Part 1

Determine the possible number of positive real zeros and negative real zeros of each polynomial function using​ Descartes' rule of signs.

f left parenthesis x right parenthesis equals 3 x Superscript 5 Baseline minus 13 x Superscript 4 Baseline plus 19 x cubed minus 5 x squared minus 9 x plus 11f(x)=3x5−13x4+19x3−5x2−9x+11

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Part 1

The possible number of positive real zeros is 

enter your response here.

To apply Descartes' Rule of Signs for determining the possible number of positive and negative real zeros of a polynomial, we examine the signs of the coefficients in two different forms of the polynomial: 

1. **For positive real zeros**: Look at \( f(x) \), the original polynomial.
2. **For negative real zeros**: Look at \( f(-x) \), by substituting \( -x \) for \( x \) and examining the sign changes.

### Step 1: Analyze \( f(x) \) for positive real zeros
The given polynomial is:

\[
f(x) = 3x^5 - 13x^4 + 19x^3 - 5x^2 - 9x + 11
\]

Now, examine the signs of each term:
- \( 3x^5 \) (positive)
- \( -13x^4 \) (negative)
- \( 19x^3 \) (positive)
- \( -5x^2 \) (negative)
- \( -9x \) (negative)
- \( 11 \) (positive)

The sign changes between consecutive terms are:
1. From \( 3x^5 \) to \( -13x^4 \) (positive to negative): 1 sign change.
2. From \( -13x^4 \) to \( 19x^3 \) (negative to positive): 1 sign change.
3. From \( 19x^3 \) to \( -5x^2 \) (positive to negative): 1 sign change.
4. From \( -5x^2 \) to \( -9x \) (negative to negative): no sign change.
5. From \( -9x \) to \( 11 \) (negative to positive): 1 sign change.

So, there are **4 sign changes** in \( f(x) \).

According to Descartes' Rule of Signs, the number of possible positive real zeros is either the number of sign changes or that number minus an even integer. Therefore, the possible number of positive real zeros is:
- 4,
- 2, or
- 0.

### Step 2: Analyze \( f(-x) \) for negative real zeros
Now, substitute \( -x \) for \( x \) in \( f(x) \):

\[
f(-x) = 3(-x)^5 - 13(-x)^4 + 19(-x)^3 - 5(-x)^2 - 9(-x) + 11
\]
Simplifying:
\[
f(-x) = -3x^5 - 13x^4 - 19x^3 - 5x^2 + 9x + 11
\]

Now, examine the signs of each term:
- \( -3x^5 \) (negative)
- \( -13x^4 \) (negative)
- \( -19x^3 \) (negative)
- \( -5x^2 \) (negative)
- \( +9x \) (positive)
- \( 11 \) (positive)

The sign changes between consecutive terms are:
1. From \( -3x^5 \) to \( -13x^4 \) (negative to negative): no sign change.
2. From \( -13x^4 \) to \( -19x^3 \) (negative to negative): no sign change.
3. From \( -19x^3 \) to \( -5x^2 \) (negative to negative): no sign change.
4. From \( -5x^2 \) to \( 9x \) (negative to positive): 1 sign change.
5. From \( 9x \) to \( 11 \) (positive to positive): no sign change.

So, there is **1 sign change** in \( f(-x) \).

According to Descartes' Rule of Signs, the number of possible negative real zeros is either the number of sign changes or that number minus an even integer. Therefore, the possible number of negative real zeros is:
- 1.

### Final Answer:
- The possible number of **positive real zeros** is 4, 2, or 0.
- The possible number of **negative real zeros** is 1.

Would you like further details or explanations? Here are some related questions to explore:

1. How does Descartes' Rule of Signs apply to polynomials with repeated factors?
2. How can we use synthetic division to confirm the number of real zeros?
3. What other methods can be used to approximate the roots of a polynomial?
4. How can we determine the number of complex roots of a polynomial?
5. How do changes in the degree of a polynomial affect the number of possible real zeros?

**Tip:** Remember that Descartes' Rule of Signs doesn't give the exact number of real zeros but only the possible numbers, so it should be used alongside other methods like synthetic division or graphing.

Determine the possible number of positive real zeros and negative real zeros of each polynomial function using Descartes' rule of signs.

f(x) = 3x^5 - 13x^4 + 19x^3 - 5x^2 - 9x + 11

Explore how to use Descartes' Rule of Signs to find the possible number of positive and negative real zeros for the polynomial 3x^5 - 13x^4 + 19x^3 - 5x^2 - 9x + 11. Learn the step-by-step process, including analyzing sign changes and applying the rule to determine real zeros. Suitable for Grades 11-12.

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