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.

Page 1: Degree and Coefficient of Polynomials

This page introduces the fundamental concepts of polynomial degree and coefficients. The degree of a polynomial is defined as the highest power of the variable (usually x) in the polynomial expression. The coefficient of a polynomial refers to the numerical factor associated with each term.

Definition: The degree of a polynomial is the largest exponent of the variable in the polynomial.

Definition: The coefficient of a polynomial is the number that multiplies the variable in each term.

The page provides several examples to illustrate these concepts:

1. W(x) = x² + 4x² + 3x² + 3x - 5 (degree 6)
2. W(x) = 8x² - x² + 10 (degree 3)
3. W(x) = √5x² + 4x + 3√5 + 1 (degree 2)

Example: In the polynomial W(x) = 8x² - x² + 10, the degree is 3, and the coefficients are 8, -1, and 10.

The page also explains how to determine coefficients when they are not explicitly stated, emphasizing that coefficients should be checked from the end of the polynomial. It's noted that terms without a variable (constant terms) have a degree of 0.

Highlight: To correctly determine the degree and coefficients of a polynomial, it should be arranged in descending order of powers.