To solve the given equation

$6x^4 - 25x^3 + 81x^2 - 9x - 13 = 0$

using Descartes' Rule of Signs, we analyze the number of possible positive and negative roots by observing the sign changes in the polynomial and its transformation.

Step 1: Analyze for Positive Roots

For positive roots, observe the polynomial $f(x) = 6x^4 - 25x^3 + 81x^2 - 9x - 13$.

• Signs of the terms: $+6, -25, +81, -9, -13$
• Count the number of sign changes:
  • $+6 \to -25$: 1 change
  • $-25 \to +81$: 1 change
  • $+81 \to -9$: 1 change
  • $-9 \to -13$: no change

Thus, there are 3 sign changes in $f(x)$. According to Descartes' Rule of Signs, the number of positive roots is either 3 or a reduced number by multiples of 2 (i.e., 3 or 1).

Step 2: Analyze for Negative Roots

To analyze for negative roots, substitute $x = -x$ in $f(x)$, giving:

$f(-x) = 6(-x)^4 - 25(-x)^3 + 81(-x)^2 - 9(-x) - 13$

Simplify:

$f(-x) = 6x^4 + 25x^3 + 81x^2 + 9x - 13$

• Signs of the terms: $+6, +25, +81, +9, -13$
• Count the number of sign changes:
  • $+6 \to +25$: no change
  • $+25 \to +81$: no change
  • $+81 \to +9$: no change
  • $+9 \to -13$: 1 change

Thus, there is 1 sign change in $f(-x)$. According to Descartes' Rule of Signs, the number of negative roots is exactly 1.

Step 3: Summary of Roots

From the analysis:

• The number of positive roots: 3 or 1
• The number of negative roots: 1
• The polynomial is of degree 4, so there are $4 - (3 \text{ or } 1 + 1) = 0 \text{ or } 2$ complex roots.

Hence, the nature of the roots is:

• Either 3 positive roots, 1 negative root, 0 complex roots, or
• 1 positive root, 1 negative root, 2 complex roots.

Would you like me to confirm this further with an explicit factorization or another method?

Related Questions:

1. How does Descartes' Rule of Signs help predict the roots of polynomials?
2. Can a fourth-degree polynomial have all real roots?
3. How can complex roots of a polynomial be determined from its degree?
4. What are the practical applications of Descartes' Rule of Signs in solving equations?
5. How does the Fundamental Theorem of Algebra relate to root prediction?

Tip: Always cross-check the results using other techniques like synthetic division or graphing for clarity.