Geometrical Meaning of Zeroes

Understanding Polynomial Graphs and Their Intersection Points

Welcome! Select a topic to explore the geometrical meaning of polynomial zeroes.
📏 Linear Polynomial
y = ax + b (a ≠ 0)
Graph: Straight line

• Intersects x-axis at exactly ONE point
• Zero: x = -b/a
• Example: y = 2x + 3, zero at x = -3/2

📊 Quadratic Polynomial
y = ax² + bx + c (a ≠ 0)
Graph: Parabola

• Opens upward if a > 0 (∪)
• Opens downward if a < 0 (∩)
• Can have 0, 1, or 2 zeroes

📈 Cubic Polynomial
y = ax³ + bx² + cx + d
Graph: S-shaped curve

• Can have 1, 2, or 3 zeroes
• Maximum 3 intersection points
• Example: x³ - 4x has 3 zeroes

⭐ Fundamental Theorem
Degree n → At most n zeroes

A polynomial of degree n can have:
• At most n real zeroes
• The graph intersects x-axis at most n times