Lecture Notes: Solving Polynomial Equations

Key Concepts Covered

• Understanding polynomial equations and methods to solve them, particularly cubic and biquadratic equations.
• The importance of rejecting alpha values in equations.
• The relationships between roots, coefficients, and equations.

Rejection of Alpha Values

• Alpha values for polynomial equations may need to be rejected based on certain conditions (e.g., roots not being satisfied).
• Example calculations for roots in the ratio of 3:4.

Example Problem: Polynomial of Degree 4

Given Polynomial

• Equation: ( x^4 - 10x^2 + 9x - 2 = 0 )
• Condition: The product of two roots is unity (e.g., if one root is ( \alpha ), the other is ( \frac{1}{\alpha} )).

Approach

• Identify relationships between roots (4 relationships can be established).
• Resulting equations may require forming a cubic equation.
• Different methods might be needed based on relationships.

Quadratic Factors

• Aim is to express a biquartic polynomial as a product of two quadratic polynomials.
• Example: Writing the quadratic factor with leading coefficient as 1, ensuring the product of two roots is unity.

Solving for Coefficients

• Use relationships from expanded equations to find values of coefficients (e.g., a, b).
• Example factorization leads to equations that need to be satisfied by a and b.

Example Problem: Another Polynomial

Given Polynomial

• Equation: ( x^4 + 12x - 5 = 0 )
• Approach similar to prior problem, but adjust assumptions based on the conditions of the polynomial.

Coefficient Calculation

• Calculate coefficients by comparing expanded forms with original polynomial.
• Use linear equations to solve for coefficients.

Final Problem: Perfect Square Polynomial

Given Polynomial

• Equation: ( x^4 - 14x^3 + 71x^2 + Px + Q )
• Condition: Determine values of P and Q for the polynomial to be a perfect square.

Steps to Solve

1. Set up the polynomial as a square of a quadratic polynomial.
2. Expand and compare coefficients.
3. Solve the resulting equations for P and Q.

Final Values

• Calculated values lead to a specific quadratic polynomial.

Summary of the Lecture

• Reviewed methods for solving cubic and biquadratic equations.
• Emphasized the relationships between coefficients and roots.
• Discussed the process of writing polynomials in specific forms.
• Homework assignment and engagement with Fermat's Last Theorem were mentioned.