Polynomial Division & Long Division Algorithm

The polynomial division involves the division of one polynomial by another. The division of polynomials can be between two monomials, a polynomial and a monomial or between two polynomials. Before learning how to divide polynomials, let’s have a brief introduction to the definition of polynomial and its related terms.

Polynomial:

A polynomial is an algebraic expression of the type a_nxⁿ + a_n−1xⁿ⁻¹+…………………a₂x² + a₁x + a₀, where “n” is either 0 or positive variables and real coefficients.

In this expression, a_n, a_n−1…..a₁, a₀ are coefficients of the terms of the polynomial.

The highest power of x in the above expression, i.e. n is known as the degree of the polynomial.

If p(x) represents a polynomial and x = k such that p(k) = 0 then k is the root of the given polynomial.

Table of Contents:

• Division of Polynomials
• Types of Polynomial Division
• Monomial by Monomial
• Polynomial by Monomial
• Polynomial by Binomial
• Polynomial by another Polynomial
• Division algorithm
• Practice problems

It is to be noted that the highest power(degree) of the polynomial gives the maximum number of zeroes of the polynomial.

Division of Polynomial

The division is the process of splitting a quantity into equal amounts. In terms of mathematics, the process of repeated subtraction or the reverse operation of multiplication is termed as division. For example, when 20 is divided by 4 we get 5 as the result since 4 is subtracted 5 times from 20.

The four basic operations viz. addition, subtraction, multiplication and division can also be performed on algebraic expressions. Let us understand the process and different methods of dividing polynomials and algebraic expressions.

Types of Polynomial Division

For dividing polynomials, generally, three cases can arise:

• Division of a monomial by another monomial
• Division of a polynomial by monomial
• Division of a polynomial by binomial
• Division of a polynomial by another polynomial

Let us discuss all these cases one by one:

Division of a monomial by another monomial

Consider the algebraic expression 40x² is to be divided by 10x then

40x²/10x = (2×2×5×2×x×x)/(2×5×x)

Here, 2, 5 and x are common in both the numerator and the denominator.

Hence, 40x²/10x = 4x

Division of a polynomial by monomial

The second case is when a polynomial is to be divided by a monomial. For dividing polynomials, each term of the polynomial is separately divided by the monomial (as described above) and the quotient of each division is added to get the result. Consider the following example:

Division of a Polynomial by Binomial

As we know, binomial is an expression with two terms. Dividing a polynomial by binomial can be done easily. Here, first we need to write the given polynomial in standard form. Now, using the long division method, we can divide the polynomial as given below.

Division of Polynomial by Another Polynomial

For dividing a polynomial with another polynomial, the polynomial is written in standard form i.e. the terms of the dividend and the divisor are arranged in decreasing order of their degrees. The method to solve these types of divisions is “Long division”. In algebra, an algorithm for dividing a polynomial by another polynomial of the same or lower degree is called polynomial long division. It is the generalised version of the familiar arithmetic technique called long division. Let us take an example.

Example: Divide  x² + 2x + 3x³ + 5 by 1 + 2x + x².

Solution:

Let us arrange the polynomial to be divided in the standard form.

3x³ + x² + 2x + 5

Divisor = x² + 2x + 1

Using the method of long division of polynomials, let us divide 3x³ + x² + 2x + 5 by x² + 2x + 1.

Step 1: To obtain the first term of the quotient, divide the highest degree term of the dividend, i.e. 3x³ by the highest degree term of the divisor, i.e. x². 

3x³/x² = 3x

Now, carry out the division process.

Step 2: Now, to obtain the second term of the quotient, divide the highest degree term of the new dividend, i.e. –5x² by the highest degree term of the divisor, i.e. x².

-5x²/x² = -5

Again carry out the division process with – 5x² – x + 5 (the remainder in the previous step).

Step 3: The remainder obtained from the previous step is 9x + 10. 

The degree of 9x + 10 is less than the divisor x² + 2x + 1. So, we cannot continue the division any further.

Polynomial Division Algorithm

If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that

p(x) = g(x) × q(x) + r(x)

Here,

r(x) = 0 or degree of r(x) < degree of g(x)

This result is called the Division Algorithm for polynomials.

From the previous example, we can verify the polynomial division algorithm as:

p(x) = 3x³ + x² + 2x + 5

g(x) = x² + 2x + 1

Also, quotient = q(x) = 3x – 1

remainder = r(x) = 9x + 10

Now,

g(x) × q(x) + r(x) = (x² + 2x + 1) × (3x – 5) + (9x + 10) 

= 3x³ + 6x² + 3x – 5x² – 10x – 5 + 9x + 10

= 3x³ + x² + 2x + 5

= p(x)

Hence, the division algorithm is verified.

Polynomial Division Questions

1. If the polynomial x⁴ – 6x³ + 16x² – 25x + 10 is divided by another polynomial x² – 2x + k, the remainder comes out to be x + a, find k and a.
2. Divide the polynomial 2t⁴ + 3t³ – 2t² – 9t – 12 by t² – 3.
3. Find all the zeroes of 2x⁴ – 3x³ – 3x² + 6x – 2, if two of its zeroes are √2 and −√2.

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