Exercise 3.1 — Zeroes of Polynomial

Identifying Coefficients, Degree, and Constant Terms

Exercise 3.1 of the Class 10 Polynomials chapter focuses on reading and analyzing the parts of a polynomial expression. Before solving problems on zeroes and factorization, students must be able to confidently identify three things in any polynomial: the coefficient of a particular term, the degree of the polynomial, and the constant term.

Consider the polynomial p(x) = 5x⁷ − 6x⁵ + 7x − 6. Here, the coefficient of x⁵ is −6, since that is the number multiplying x⁵. The degree of the polynomial is 7, because that is the highest power of x present in the expression. The constant term is −6, which is the term that does not contain any variable. Practicing this kind of identification builds the foundation needed for more advanced operations on polynomials.

True or False: Testing Conceptual Understanding

This exercise also includes true-or-false style questions that test whether students truly understand the definitions of degree, coefficient, and polynomial. These questions are especially useful for CBSE, Telangana, and Andhra Pradesh board exams, where conceptual statements are often asked in multiple-choice or short-answer format.

• Degree of a constant term is always zero — true, because a constant has no variable, so its variable power is considered zero
• Degree equals the number of terms plus one — false, since degree depends only on the highest power of the variable, not the number of terms
• Coefficient of x² in 3x³ − 4x² + 5x + 7 — the correct coefficient is −4, not any other number, so always read the sign carefully
• Expressions with the variable in the denominator — such as 1/(x² − 5x + 6), are not polynomials at all, regardless of their appearance

Evaluating a Cubic Polynomial

A major part of this exercise involves finding the value of a polynomial at different points by substitution. For the cubic polynomial p(t) = t³ − 1, students substitute various values of t and simplify step by step.

For example, p(1) = 1³ − 1 = 0, p(−1) = (−1)³ − 1 = −2, p(0) = 0³ − 1 = −1, p(2) = 2³ − 1 = 7, and p(−2) = (−2)³ − 1 = −9. Notice how the cube of a negative number remains negative, which is a common point of confusion. Careful attention to signs while cubing negative numbers is essential to avoid errors in this type of question.

Worked Example: Verifying Zeroes of a Biquadratic Polynomial

Exercise 3.1 also asks students to check whether given numbers are zeroes of a polynomial by substitution. For the polynomial p(x) = x⁴ − 16, substituting x = −2 gives p(−2) = (−2)⁴ − 16 = 16 − 16 = 0, and substituting x = 2 gives p(2) = 2⁴ − 16 = 16 − 16 = 0.

Since both substitutions result in zero, both −2 and 2 are confirmed as zeroes of this biquadratic polynomial. This verification method — substitute, simplify, and check if the result equals zero — is the standard approach used throughout the chapter, and it also forms the basis for the factor theorem studied later.

Worked Example: Verifying Zeroes of a Quadratic Polynomial

For the quadratic polynomial p(x) = x² − x − 6, students check whether 3 and −2 are zeroes. Substituting x = 3 gives p(3) = 3² − 3 − 6 = 9 − 9 = 0, and substituting x = −2 gives p(−2) = (−2)² − (−2) − 6 = 4 + 2 − 6 = 0. Since both results equal zero, 3 and −2 are confirmed as the zeroes of this polynomial.

This example reinforces an important pattern: a quadratic polynomial generally has exactly two zeroes, and verifying them by direct substitution is a reliable method before moving on to factorization techniques.

Common Mistakes to Avoid

• Mixing up the coefficient and the constant term — the coefficient belongs to a variable term, while the constant term has no variable
• Forgetting sign rules when raising negative numbers to odd or even powers
• Assuming the degree is related to the number of terms instead of the highest exponent
• Treating expressions with variables in the denominator as polynomials
• Making arithmetic slips while substituting values into higher-degree polynomials such as x⁴ or x³

What This Lesson Prepares You For

Mastering coefficients, degrees, and zero-verification in this exercise prepares students for the next stage of the chapter, where the relationship between the zeroes and coefficients of a polynomial is studied in detail, along with the division algorithm for polynomials. These skills are also directly useful while solving quadratic equations later in the syllabus.

For a refresher on the basic definitions used throughout this chapter, revisit the introduction to polynomials, which covers degree, value, and zeroes in detail with simple examples.