Zeroes of a Polynomial

What is a Zero?

A zero (or root) of polynomial $p(x)$ is a value $k$ such that $p(k) = 0$ .

In other words: when you plug in $k$ for $x$ , you get 0!

Finding Zeroes: Examples

Example 1: $p(x) = x - 3$

Step 1: Set $p(x) = 0$ $x - 3 = 0$

Step 2: Solve $x = 3$

Step 3: Verify $p(3) = 3 - 3 = 0$ ✓

Zero: $x = 3$

Example 2: $p(x) = x^2 - 9$

Step 1: Set $p(x) = 0$ $x^2 - 9 = 0$

Step 2: Factor $(x + 3)(x - 3) = 0$

Step 3: Solve each factor $x + 3 = 0 \Rightarrow x = -3$ $x - 3 = 0 \Rightarrow x = 3$

Zeroes: $x = -3$ and $x = 3$

Example 3: $p(x) = 2x + 5$

Step 1: Set $p(x) = 0$ $2x + 5 = 0$

Step 2: Solve $2x = -5$ $x = -\frac{5}{2}$

Zero: $x = -\frac{5}{2}$

Maximum Number of Zeroes

A polynomial of degree $n$ has at most $n$ zeroes.

Degree	Max Zeroes	Example
1 (Linear)	1	$x - 5 = 0$ → one zero
2 (Quadratic)	2	$x^2 - 9 = 0$ → two zeroes
3 (Cubic)	3	Could have up to 3 zeroes

Class 9 Maths (NCERT)

Module 1: Chapter 1: Number Systems

Module 2: Chapter 2: Polynomials

Module 3: Chapter 3: Coordinate Geometry

Module 4: Chapter 4: Linear Equations in Two Variables

Module 5: Chapter 5: Introduction to Euclid's Geometry

Module 6: Chapter 6: Lines and Angles

Module 7: Chapter 7: Triangles

Module 8: Chapter 8: Quadrilaterals

Module 9: Chapter 9: Circles

Module 10: Chapter 10: Heron's Formula

Module 11: Chapter 11: Surface Areas and Volumes

Module 12: Chapter 12: Statistics

Module 13: Chapter 13: Constructions

Module 14: Chapter 14: Probability

Chapter 2: Polynomials

What is a Zero?

Finding Zeroes: Examples

Example 1: $p(x) = x - 3$

Example 2: $p(x) = x^2 - 9$

Example 3: $p(x) = 2x + 5$

Maximum Number of Zeroes

Class 9 Maths (NCERT)

Module 1: Chapter 1: Number Systems

Module 2: Chapter 2: Polynomials

Module 3: Chapter 3: Coordinate Geometry

Module 4: Chapter 4: Linear Equations in Two Variables

Module 5: Chapter 5: Introduction to Euclid's Geometry

Module 6: Chapter 6: Lines and Angles

Module 7: Chapter 7: Triangles

Module 8: Chapter 8: Quadrilaterals

Module 9: Chapter 9: Circles

Module 10: Chapter 10: Heron's Formula

Module 11: Chapter 11: Surface Areas and Volumes

Module 12: Chapter 12: Statistics

Module 13: Chapter 13: Constructions

Module 14: Chapter 14: Probability

Chapter 2: Polynomials

Zeroes of a Polynomial

What is a Zero?

Finding Zeroes: Examples

Example 1: p(x)=x−3p(x) = x - 3p(x)=x−3

Example 2: p(x)=x2−9p(x) = x^2 - 9p(x)=x2−9

Example 3: p(x)=2x+5p(x) = 2x + 5p(x)=2x+5

Maximum Number of Zeroes

Example 1: $p(x) = x - 3$

Example 2: $p(x) = x^2 - 9$

Example 3: $p(x) = 2x + 5$