Walkthrough Examples: Applying Descartes' Rule of Signs to Polynomials

📚 Quick Study Guide

• ➕ Descartes' Rule of Signs helps determine the possible number of positive and negative real roots of a polynomial.
• 🔢 To find the possible number of positive real roots, count the number of sign changes in $f(x)$. The number of positive real roots is equal to this number or less than this number by an even integer.
• ➖ To find the possible number of negative real roots, count the number of sign changes in $f(-x)$. The number of negative real roots is equal to this number or less than this number by an even integer.
• 💡 Remember to include multiplicity when considering the number of roots.
• ✍️ Missing terms in the polynomial should be treated as having a coefficient of 0 when counting sign changes.

📝 Practice Quiz

1. Question 1: How many possible positive real roots does the polynomial $f(x) = x^3 - 2x^2 + x - 5$ have?
   1. 0 or 2
   2. 1 or 3
   3. 1
   4. 2
2. Question 2: How many possible negative real roots does the polynomial $f(x) = x^5 + 3x^4 - x^3 + 2x^2 - x + 7$ have?
   1. 0
   2. 2 or 0
   3. 1 or 3
   4. 1
3. Question 3: Determine the possible number of positive real roots for $f(x) = 2x^4 - x^3 + 5x^2 + 3x - 8$.
   1. 0 or 2
   2. 1 or 3
   3. 2 or 0
   4. 1
4. Question 4: Determine the possible number of negative real roots for $f(x) = x^6 - 4x^4 + 3x^2 - 12$.
   1. 0
   2. 2
   3. 1 or 3
   4. 0, 2, or 4
5. Question 5: Find the possible number of positive real roots of the polynomial $f(x) = x^4 + 5x^2 + 4$.
   1. 0
   2. 2
   3. 4
   4. 1
6. Question 6: What are the possible numbers of negative real roots for the polynomial $f(x) = x^3 - 7x - 6$?
   1. 0
   2. 1
   3. 2
   4. 3
7. Question 7: Determine the possible number of positive real roots for $f(x) = x^5 - 3x^3 + 2x - 1$.
   1. 1 or 3
   2. 1
   3. 3 or 1 or 5
   4. 2