1. Introduction

A polynomial is an expression consisting of variables and coefficients, involving operations like addition, subtraction, multiplication, and non-negative integer exponents.

Examples:

• [math] 4x + 2 [/math]: Degree 1
• [math] 5x^3 – 4x^2 + x – 2 [/math]: Degree 3

Types of Polynomials by Degree:

• Linear Polynomial: [math] ax + b [/math] (Degree 1)
• Quadratic Polynomial: [math] ax^2 + bx + c [/math] (Degree 2)
• Cubic Polynomial: [math] ax^3 + bx^2 + cx + d [/math] (Degree 3)

2. Zeroes of a Polynomial

A zero of a polynomial [math] p(x) [/math] is a value [math] k [/math] such that [math] p(k) = 0 [/math].

Example:

• For [math] p(x) = x^2 – 3x – 4 [/math]:
• [math] p(-1) = 0 [/math], [math] p(4) = 0 [/math]
• Zeroes: [math] -1 [/math] and [math] 4 [/math]

3. Geometrical Meaning of Zeroes

For a linear polynomial ([math] y = ax + b [/math]), the graph is a straight line intersecting the x-axis at one point, which is its zero.

For a quadratic polynomial ([math] y = ax^2 + bx + c [/math]):

• Two distinct zeroes: The parabola intersects the x-axis at two points.
• One zero (double root): The parabola touches the x-axis.
• No real zeroes: The parabola lies entirely above or below the x-axis.