Exercise 2.2 — Zeroes of a Polynomial

Understanding Zeroes of a Polynomial

In Class 9 Mathematics Chapter 2, Polynomials and Factorisation, one of the most important ideas is the concept of a zero of a polynomial. If p(x) is a polynomial, then the value of x for which p(x) = 0 is called a zero of that polynomial. This value is also known as the root of the equation p(x) = 0. Exercise 2.2 builds on this idea through a series of problems that ask students to evaluate polynomials at given points, verify whether certain numbers are zeroes, and find zeroes of linear polynomials. These skills form the foundation for factorisation and solving polynomial equations in later chapters.

Finding the Value of a Polynomial at a Given Point

The first set of questions asks students to substitute given values of x into a polynomial and simplify. For example, to evaluate p(x) = 4x² − 5x + 3 at x = −1, simply replace every x with −1 and simplify carefully, paying close attention to signs.

This kind of substitution is repeated for fractional values such as x = 1/2, where students must work confidently with fractions and exponents. Mastering this step is essential because evaluating p(a) correctly is the basis for checking whether a number is a zero of the polynomial.

Verifying Whether a Given Value is a Zero

A large part of Exercise 2.2 focuses on checking whether a particular value of x makes the polynomial equal to zero. The method is simple: substitute the value into p(x) and simplify.

• If p(a) = 0, then x = a is a zero of the polynomial.
• If p(a) ≠ 0, then x = a is not a zero of the polynomial.
• For polynomials like p(x) = x² − 1, both x = 1 and x = −1 can be zeroes, since a quadratic polynomial can have up to two zeroes.
• For products like p(x) = (x − 1)(x + 2), a value makes p(x) = 0 only if it makes one of the factors zero.

This question type strengthens algebraic manipulation skills, especially when dealing with negative numbers and fractions such as −1/3 or 2/3, which are common in CBSE, Telangana, and Andhra Pradesh board exams.

Finding the Zero of a Linear Polynomial

The next part of the exercise teaches a general method for finding the zero of a linear polynomial. For a polynomial of the form f(x) = ax + b, the zero is found by setting the polynomial equal to zero and solving for x.

Using this rule, students can quickly find zeroes of polynomials such as f(x) = x + 2 (zero at x = −2), f(x) = 2x + 3 (zero at x = −3/2), and f(x) = px + q (zero at x = −q/p). Remembering this general formula saves time and avoids repeated step-by-step solving during exams.

Using Zeroes to Find Unknown Coefficients

The final problems in Exercise 2.2 reverse the process: instead of finding the zero, students are given a zero and must find an unknown coefficient in the polynomial. For example, if x = 2 is a zero of p(x) = 2x² − 3x + 7a, then p(2) = 0 can be used to set up and solve an equation for a.

Similarly, when two zeroes of a cubic polynomial are given, both conditions are used together to form a system of equations and find the unknown constants. This step is an important bridge to the next topics, factorisation and the factor theorem.

Common Mistakes to Avoid

• Forgetting to apply the correct sign when substituting negative values into squares and cubes.
• Mixing up "value of the polynomial" with "zero of the polynomial" — these are different things.
• Errors while simplifying fractions, especially with exponents like (1/2)² or (−1/3)².
• Not checking the final answer by substituting it back into the original polynomial.

What This Lesson Prepares You For

A strong grasp of zeroes of a polynomial directly supports the next sections of this chapter, including factorisation using the factor theorem and splitting the middle term. Students preparing for CBSE, Telangana, or Andhra Pradesh board exams should also revisit related topics such as Introduction to Polynomials and Introduction to Factorisation to build a complete understanding before attempting Exercise 2.3.