Class 9 - Finding Zeroes of a Polynomial

Zeroes of a polynomial is the value of the variable that makes the polynomial zero. Finding the roots or zeroes of a polynomial is one of the important topics covered under algebra as it forms the basis of solving polynomial equations. On this page, we will cover the topic of zeroes of polynomials in detail along with definition, methods, formulas and examples based on finding zeroes of a polynomial.

Table of Contents

• What are Zeroes of Polynomial
• How to Find Zeroes of Polynomial
• Solved Examples on Finding the zeroes of a Polynomial

What are Zeroes of Polynomial

Zeroes of a polynomial also known as the roots of a polynomial refer to the value of the variable for which the polynomial becomes zero.

Example: If p(x) denotes a polynomial 5x3 – 2x2 + 3x – 2.

Then if we replace x by 1 everywhere in p(x), we get:

• p(1) = 5 × (1)3 – 2 × (1)3 + 3 × (1) – 2 = 5 – 2 + 3 – 2 = 4
• Similarly, we can say that 1 is the zero of x – 1, and –2 is the zero of x + 2.

How to Find Zeroes of Polynomial

Finding a zero of the polynomial p(x), means solving the polynomial equation such that p(x) = 0. If p(x) = ax + b, a ≠ 0, is a linear polynomial, we can find a zero of p(x) by equating it to zero and solving for the value of variable:

• So, first write p(x) = 0 means ax + b = 0, a ≠ 0 So, ax = –b
• i.e., x = –b/a
• So, x = –b/a is the only zero of p(x), i.e., A polynomial has one and only one zero.

Facts About Zeros of a Polynomial:

• (i) A zero of a polynomial need not be 0.
• (ii) 0 may be a zero of a polynomial.
• (iii) Every linear polynomial has one and only one zero.
• (iv) A polynomial can have more than one zero.

Solved Examples on Finding the zeroes of a Polynomial

Example 1: Find a zero of the polynomial p(x) = 2x + 1.

Solution: Finding a zero of p(x), is the same as solving the equation. I.e., equating p(x) to zero: p(x) = 0

Now substitute the value of p(x), 2x + 1 = 0

This gives us x = –1/2

So, –1/2 is a zero of the polynomial 2x + 1.

Example 2: Check whether –2 and 2 are zeroes of the polynomial x + 2.

Solution: Let p(x) = x + 2.

Then p(2) = 2 + 2 = 4, p(–2) = –2 + 2 = 0

Therefore, –2 is a zero of the polynomial x + 2, but 2 is not.

Example 3: Find a zero of the polynomial p(x) = x + 1.

Solution: Finding a zero of p(x), is the same as solving the equation

p(x) = 0

Now, x + 1 = 0 gives us x = – 1

So, –1 is a zero of the polynomial x + 1.

Example 4: Verify whether 2 and 0 are zeroes of the polynomial x2 – 2x.

Solution: Given polynomials p(x) = x2– 2x

Then p(2) = 22– 4 = 4 – 4 = 0

and p(0) = 0 – 0 = 0

Hence, 2 and 0 are both zeroes of the polynomial x2– 2x.