IOQM Lecture Notes

Introduction

• Focus on previous questions from PRMO and RMO.
• Discussing Rational Root Theorem and Integer Root Theorem.

Rational Root Theorem

• If p/q is a root of a polynomial equation:
  • q must be a factor of the leading coefficient (a_n).
  • p must be a factor of the constant term (a_0).
• p and q should be co-prime (GCD(p, q) = 1).

Example Polynomial

• Consider the quadratic equation: 6x² - 5x + 1 = 0.
• Factoring gives roots: x1 = 1/2 and x2 = 1/3.

Integer Root Theorem

• Roots are integers if the leading coefficient (a) is 1.
• If a is 1, p/q will also be an integer.

Application Example

• Given: x² + ax + 20 = 0 (with integer roots).
• Sum of roots: -a = x1 + x2.
• Product of roots: 20 = x1 * x2.
• Possible pairs for (x1, x2): 20 & 1, 10 & 2, 5 & 4.
• Calculate possible values for a based on pairs:
  • From pairs: (20, 1): Sum is 21 -> a = -21.
  • From pairs: (10, 2): Sum is 12 -> a = -12.
  • From pairs: (5, 4): Sum is 9 -> a = -9.
• Total sum of possible a values = 0.*

Further Topics

• Integer Root Theorem again explained with examples.
• Discussing the transformation of roots and its implications for functions.

Complex Numbers and Equations

• Expression involving x/y + y/z + z/x, leading to rational outputs.
• Solve for maximum/minimum values using discriminants and properties of polynomials.

Important Concepts

1. Transformation of Roots:
   • If roots are altered (like x -> -x), corresponding equations change.
2. Discriminants:
   • Used to determine the nature of roots in quadratic equations.
3. Quadratic Equations:
   • Familiarity with completing the square and applying formulas is crucial.

Example Problems

• Solve for minimum and maximum of expressions with rational roots.
• Given expressions lead to systems of equations.
• Applications of roots in determining values of a, b, c through algebraic manipulation.

Conclusion

• Summary of the key concepts learned in this class.
• Encouragement to practice previous years' questions to strengthen understanding.
• Homework: Solve assigned problems for reinforcement.