Page 2: Polynomial Exercises and Problem Solving

This page focuses on practical exercises involving polynomials, demonstrating how to apply the concepts learned on the previous page. The exercises cover various aspects of polynomial manipulation and analysis.

The first exercise involves finding the coefficient of a polynomial given specific conditions:

Example: Find the coefficient 'a' in the polynomial W(x) = -x⁴ - 2x³ + ax + 3 if W(-2) = -1.

The solution involves substituting -2 for x and solving the resulting equation to find the value of 'a'.

The second exercise requires determining two coefficients of a polynomial based on given conditions:

Example: Find coefficients 'a' and 'b' in the polynomial W(x) = 2x² + ax² + x + b such that W(1) = -5 and W(-1) = -1.

This problem is solved by creating a system of equations using the given conditions and solving for 'a' and 'b'.

The third exercise demonstrates a more complex problem involving irrational numbers:

Example: Determine the coefficient 'a' in the polynomial W(x) = x² + ax - 4 if W(√2 - 1) = 4(1 - √3).

This problem requires careful algebraic manipulation and simplification of irrational expressions.

Highlight: These exercises demonstrate the importance of algebraic manipulation skills and the ability to apply polynomial concepts to solve complex problems.

The page concludes with detailed solutions for each exercise, providing step-by-step explanations of the problem-solving process. This approach helps reinforce the concepts and techniques learned, preparing students for more advanced polynomial problems.