Exercise 2.3 — Remainder Theorem

Division of Polynomials and the Remainder Theorem

Exercise 2.3 of Class 9 Mathematics Chapter 2, Polynomials and Factorisation, introduces one of the most powerful tools for working with polynomials — the Remainder Theorem. Just as numbers can be divided to get a quotient and remainder, polynomials can also be divided by other polynomials. This exercise shows students a shortcut to find the remainder without performing long division every time, which saves valuable time during CBSE, Telangana, and Andhra Pradesh board exams.

Recalling Division: Dividend, Divisor, Quotient, and Remainder

Before applying the Remainder Theorem, it helps to recall the basic division relationship used for numbers and polynomials alike:

When a polynomial p(x) is divided by another polynomial g(x), the same relationship holds: p(x) = g(x) · q(x) + r(x), where the degree of the remainder r(x) is always less than the degree of the divisor g(x). If g(x) is a linear polynomial like x − a, then r(x) must be a constant — and this constant is exactly what the Remainder Theorem helps us find quickly.

Statement of the Remainder Theorem

The Remainder Theorem states that if a polynomial p(x) of degree one or more is divided by the linear polynomial (x − a), then the remainder is simply p(a) — the value of the polynomial at x = a.

This means students no longer need to perform full polynomial long division just to find the remainder. Instead, they identify the zero of the divisor and substitute it into the original polynomial. For example, dividing p(x) = x² − 5x + 6 by g(x) = x − 3 gives p(3) = 9 − 15 + 6 = 0. Since the remainder is zero, this also tells us that x − 3 is a factor of p(x) — a key idea that leads directly into the Factor Theorem in the next section.

Applying the Theorem: Finding the Zero of the Divisor

The most important skill in Exercise 2.3 is correctly identifying the zero of the divisor before substituting. For a divisor like x + 1, the zero is x = −1; for 2x − 3, the zero is x = 3/2; and for 5 + 2x, the zero is x = −5/2. Once the correct value is found, it is substituted into p(x) and simplified carefully — especially when dealing with negative numbers, fractions, and irrational numbers like π.

• Linear divisors such as x − a, ax + b, or px − q always have a single zero, found by setting the divisor equal to zero.
• Sign errors are the most common mistake — always double-check whether the zero is positive or negative before substituting.
• Fractional and irrational zeroes require careful simplification using exponent rules.

Worked Example: Two Polynomials with Equal Remainders

Some problems in this exercise involve two different polynomials that leave the same remainder when divided by the same linear divisor. The approach is to apply the Remainder Theorem to both polynomials separately, set the two remainders equal to each other, and solve the resulting equation for the unknown coefficient.

For instance, if f(x) = 2x³ + ax² + 3x − 5 and g(x) = x³ + x² − 4x + a leave the same remainder when divided by x − 2, then f(2) = g(2). Substituting x = 2 into both expressions and simplifying gives a simple linear equation in a, which can be solved step by step. This type of question is frequently asked in board exams because it combines substitution skills with solving linear equations.

Verifying the Remainder Theorem by Long Division

To build confidence in the theorem, some questions ask students to verify the remainder using actual polynomial long division. For example, dividing f(x) = x⁴ − 3x² + 4 by g(x) = x − 2 using long division gives a remainder of 8, which exactly matches the value of f(2) calculated using the Remainder Theorem. This verification step is important for understanding — once students trust the theorem, long division is rarely needed for linear divisors.

What This Lesson Prepares You For

The Remainder Theorem is the stepping stone to the Factor Theorem, which is used extensively in factorising cubic and higher-degree polynomials. Students preparing for CBSE, Telangana, and Andhra Pradesh board exams should revisit Exercise 2.2 on Zeroes of a Polynomial to strengthen the substitution skills used here, and continue with Introduction to Factorisation to see how the remainder theorem connects directly to factorising polynomials.