Division Algorithm for Polynomial

The division algorithm for polynomials is a method for dividing one polynomial by another. It is similar to the long division algorithm for integers but with some key differences. We can perform the division of polynomials using various steps that are,

Step 1: The first step is to write the dividend and divisor in decreasing order of their degrees. The dividend is the polynomial that is being divided, and the divisor is the polynomial that is doing the dividing.

Step 2: Next, we need to find the first term of the quotient. This is done by dividing the leading term of the dividend by the leading term of the divisor. The result is the first term of the quotient, and it is placed above the bar.

Step 3: Now, we need to subtract the product of the first term of the quotient and the divisor from the dividend. This gives us a new polynomial, which is called the remainder.

Step 4: If the degree of the remainder is less than the degree of the divisor, then we are finished. The quotient is the polynomial with the first term that we found, and the remainder is the polynomial that we just subtracted.

Step 5: If the degree of the remainder is equal to the degree of the divisor, then we need to repeat the process. We find the next term of the quotient by dividing the leading term of the remainder by the leading term of the divisor. We then subtract the product of the new term of the quotient and the divisor from the remainder. We continue this process until the degree of the remainder is less than the degree of the divisor.

The division algorithm for polynomials can be used to solve a variety of problems, such as finding the roots of a polynomial and finding the inverse of a polynomial.

Example: Divide the polynomial f(x) = 3x² − x³ − 3x + 5 by the polynomial g(x) = x − 1 − x² and verify the division algorithm.

Solution:

Rewrite the given polynomial in standard form.

f(x) = -x³ + 3x² – 3x + 5 and g(x) = -x² + x – 1

Use the long division method,

 

So, the remainder r(x) is 3.

Quotient × Divisor + Remainder = (x − 2)(−x² + x − 1) + 3

−x³ + x² − x + 2x² −2x +2 + 3

−x³ + 3x² − 3x + 5  (Dividend)

Hence, the division algorithm, is verified.

CBSE Class 10 Maths Notes Chapter 4 Polynomials are provided to support student’s education. We believe that students’ learning and development are of utmost importance, and that’s why we have created these comprehensive notes to help them comprehend the complex subject of Polynomials better.

The NCERT Class 10 Maths textbook’s Chapter 4 explores the realm of Polynomials and covers various concepts such as finding the polynomial degree, types of polynomials, zeros of polynomials, and more. Our notes aim to provide students with a complete summary of the entire chapter, including all essential topics, formulae, and concepts necessary to succeed in their exams.