Step 1: Descartes's Rule of Signs

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

Function:

$f(x) = 2x^4 + 3x^3 - 9x^2 - 17x - 6$

Number of Possible Positive Zeros:

To find the possible number of positive real zeros, we check the sign changes in $f(x)$.

1. The signs of the terms in $f(x)$ are: $2x^4 \quad (+), \quad 3x^3 \quad (+), \quad -9x^2 \quad (-), \quad -17x \quad (-), \quad -6 \quad (-)$

2. The sign changes are between:

   • From $3x^3$ to $-9x^2$ (positive to negative)

Thus, there is one sign change. According to Descartes’s rule of signs, the number of possible positive zeros is 1.

Number of Possible Negative Zeros:

To find the possible number of negative real zeros, we substitute $-x$ into $f(x)$ and analyze the sign changes in $f(-x)$.

Substitute $-x$ into $f(x)$: $f(-x) = 2(-x)^4 + 3(-x)^3 - 9(-x)^2 - 17(-x) - 6$ Simplifying: $f(-x) = 2x^4 - 3x^3 - 9x^2 + 17x - 6$

Now, the signs of the terms in $f(-x)$ are: $2x^4 \quad (+), \quad -3x^3 \quad (-), \quad -9x^2 \quad (-), \quad +17x \quad (+), \quad -6 \quad (-)$

The sign changes are between:

• From $2x^4$ to $-3x^3$ (positive to negative)
• From $+17x$ to $-6$ (positive to negative)

Thus, there are two sign changes in $f(-x)$. According to Descartes's rule of signs, the possible number of negative zeros is 2 or 0 (since the number of real zeros decreases by even numbers).

Final Conclusion:

• Possible number of positive real zeros: 1
• Possible number of negative real zeros: 2 or 0

Do you need further clarification on any step or details?

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Here are five related questions:

1. How does Descartes's rule of signs apply to more complex polynomials with higher degrees?
2. What happens if there are no sign changes in $f(x)$?
3. How can we verify the actual number of real zeros after finding the possible number using Descartes’s rule of signs?
4. Does Descartes's rule give information about complex (non-real) zeros?
5. What additional methods can be used to precisely locate the zeros of a polynomial?

Tip:

Descartes’s rule only tells you the possible number of real zeros—it doesn’t guarantee how many real zeros there are, so further analysis or graphing may be needed.