Exercise 3.4 — Division Algorithm

Understanding the Division Algorithm for Polynomials

Exercise 3.4 of Class 10 Polynomials introduces the division algorithm, a method that extends the familiar process of dividing numbers to dividing one polynomial by another. This concept is heavily tested in CBSE, Telangana, and Andhra Pradesh board exams because it links directly to factorisation, finding unknown zeroes, and verifying relationships between polynomials.

The division algorithm states that for any dividend polynomial p(x) and a non-zero divisor polynomial g(x), there exist a unique quotient q(x) and remainder r(x) such that the following relationship always holds true.

Here, r(x) is either zero, or its degree is strictly less than the degree of g(x). This single rule forms the basis for every problem in this exercise.

Key Rules to Remember

Before attempting division problems, students should be familiar with a few important facts that connect the degrees of these polynomials and explain what happens in special cases.

• Degree relationship — if g(x) has degree 1, then the degree of p(x) equals 1 plus the degree of q(x).
• Remainder Theorem — if p(x) is divided by (x − a), the remainder equals p(a).
• Factor condition — if r(x) = 0, then g(x) divides p(x) exactly, meaning both g(x) and q(x) are factors of p(x).

Step-by-Step Method for Dividing Polynomials

The long division method for polynomials follows the same logic as numerical long division, repeated until the remainder has a smaller degree than the divisor.

• Arrange both the dividend and divisor in standard form, from the highest degree term to the lowest, filling in any missing powers with zero coefficients.
• Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient.
• Multiply this quotient term by the entire divisor and subtract the result from the dividend.
• Repeat the process with the new remainder until its degree becomes less than the degree of the divisor.

For instance, dividing x³ − 3x² + 5x − 3 by x² − 2 gives a quotient of x − 3 with a remainder of 7x − 9, since the remainder's degree (1) is less than the divisor's degree (2), satisfying the division algorithm condition.

Checking if One Polynomial is a Factor of Another

A common application of this exercise is checking whether one polynomial completely divides another. If the division leaves a remainder of zero, the divisor is a factor of the dividend. For example, dividing 2t⁴ + 3t³ − 2t² − 9t − 12 by t² − 3 leaves a remainder of zero, confirming that t² − 3 is a factor of the larger polynomial. On the other hand, if division leaves a nonzero remainder, the first polynomial is not a factor of the second — this distinction is frequently asked as a direct exam question.

Finding All Zeroes Using Known Zeroes

Some questions provide a few zeroes of a higher-degree polynomial and ask students to find the remaining ones. The method involves writing factors corresponding to the known zeroes, multiplying them together to get a quadratic factor, and then dividing the original polynomial by this factor. The quotient obtained is itself a polynomial whose zeroes complete the full set of zeroes for the original polynomial. This technique combines the division algorithm with factorisation skills learned in earlier exercises.

Common Mistakes to Avoid

• Forgetting to write missing terms with a zero coefficient when arranging the dividend in standard form.
• Making sign errors while subtracting during each step of the division.
• Stopping the division too early or too late — always check that the remainder's degree is less than the divisor's degree.
• Confusing the quotient and remainder when applying the division algorithm formula.

What This Lesson Prepares You For

The division algorithm is a crucial tool that connects back to Exercise 3.3 on the relationship between zeroes and coefficients, since both rely on factorisation and understanding how zeroes relate to a polynomial's structure. It also lays the groundwork for solving Quadratic Equations, where dividing and factoring expressions is a regular requirement. Students moving from Class 9 Polynomials will find that this exercise builds confidence in handling higher-degree expressions, a skill that proves valuable throughout board exam preparation in CBSE, Telangana, and Andhra Pradesh syllabi.