Division Algorithm

Division is an arithmetic operation that involves grouping objects into equal parts. It is also understood as the inverse operation of multiplication. For example, in multiplication, 3 groups of 6 make 18. Now, if 18 is divided into 3 groups, it gives 6 objects in each group. Here 18 is the dividend, 3 is the divisor and 6 is the quotient. The dividend is the product of the divisor and the quotient, added to the remainder (if any) and this rule is known as the division algorithm. The division algorithm applies to the division of polynomials as well.

The division of polynomials involves dividing one polynomial by a monomial, binomial, trinomial, or a polynomial of a lower degree. In a polynomial division, the degree of the dividend is greater than or equal to the divisor. To verify the result, we multiply the divisor polynomial and the quotient and add it to the remainder, if any. i.e., we use the division algorithm to verify the result.

The division algorithm says when a number 'a' is divided by a number 'b' gives the quotient to be 'q' and the remainder to be 'r' then a = bq + r where 0 ≤ r < b. This is also known as "Euclid's division lemma". The division algorithm can be represented in simple words as follows:

• Dividend = Divisor × Quotient + Remainder

Let us just verify the division algorithm for some numbers. We know that when 59 is divided by 7, the quotient is 8 and the remainder is 3. Here,

• dividend = 59
• divisor = 7
• quotient = 8
• remainder = 3
• Verification of division algorithm:
  Dividend = Divisor × Quotient + Remainder
  59 = 7 × 8 + 3
  59 = 56 + 3
  59 = 59
  Hence, the division algorithm is verified.

Here is another example of division algorithm.

The division algorithm for polynomials says, if p(x) and g(x) are the two polynomials, where g(x) ≠ 0, we can write the division of polynomials as: p(x) = q(x) × g(x) + r(x), where the degree of r(x) < degree of g(x) and

• p(x) is the dividend
• g(x) is the divisor
• q(x) is the quotient
• r(x) is the remainder

If we compare this to the regular division of numbers, we can easily understand this as: Dividend = (Divisor × Quotient) + Remainder. We will verify the division algorithm for polynomials in the following example.

Example: Find the quotient and the remainder when the polynomial 4x³ + 5x² + 5x + 8 is divided by (4x + 1) and verify the result by the division algorithm.

Solution:

First, we divide the given polynomial p(x) = 4x³ + 5x² + 5x + 8 by g(x) = (4x + 1) using long division.

We found the quotient to be q(x) = x² + x + 1 and r(x) = 7. We will now verify the division algorithm.

p(x) = q(x) × g(x) + r(x)

4x³ + 5x² + 5x + 8 = (x² + x + 1) (4x + 1) + 7

4x³ + 5x² + 5x + 8 = 4x³ + 4x² + 4x + x² + x + 1 + 7

4x³ + 5x² + 5x + 8 = = 4x³ + 5x² + 5x + 8

Thus, the division algorithm is verified.

The steps for the polynomial division are given below.

Step 1: Arrange the dividend and the divisor in the descending order of their exponents.

Step 2: Find the first term of the quotient by dividing the highest degree term of the dividend by the highest degree term of the divisor.

Step 3: Then multiply the divisor by the current quotient and subtract the result from the current dividend. This will give a new dividend.

Step 4: Find the next term of the quotient by dividing the largest degree term of the new dividend obtained in step 3 by the largest degree term of the divisor.

Step 5: Repeat steps 3 and 4 again until the degree of the remainder is less than the degree of the divisor.

Let us understand this process with an example: Divide 2x³ + 3x² + 4x + 3 by x + 1.

Here, p(x) = 2x³ + 3x² + 4x + 3 and g(x) = x + 1. We will use the above steps to divide p(x) by g(x).

Step 1 : The polynomials are already arranged in the descending order of their degrees.

Step 2: The first term of the quotient is obtained by dividing the largest degree term of the dividend with the largest degree term of the divisor.

∴ First term = (2x³) / x = 2x².

Step 3: Then the new dividend is x² + 4x which is obtained as follows:

Step 4: The second term of the quotient is obtained by dividing the largest degree term of the new dividend obtained in step 2 with the largest degree term of the divisor.

Second term = (x²)/x = x.

Step 5: Repeat steps 3 and 4 again until the remainder's degree is less than the divisor's degree. Then we get the quotient to be 2x² + x + 3.

Here p(x) = 2x³ + 3x² + 4x + 3, g(x) = x + 1, q(x) = 2x² + x + 3 and r(x) = 0. Try verifying the division algorithm for polynomials now.

Division Algorithm For Linear Divisors

When a polynomial of degree n ≥ 1 is divided by a divisor with degree 1, then we call it as a division by linear divisor. The division algorithm for linear divisors is the same as that of the polynomial division algorithm discussed above except for the fact that the divisor is of degree 1.

Let us look at an example below: Let p(x) = x² + x + 1 be the dividend and g(x) = x − 1 be the divisor. Here the degree of the divisor is 1. Here g(x) is called a "Linear divisor". To know more about this division algorithm, please click here. Let us divide p(x) by g(x).

Let's verify the division algorithm for polynomials here.

x² + x + 1 = (x - 1) (x + 2) + 3

x² + x + 1 = x² + 2x - 1x - 2 + 3

x² + x + 1 = x² + x + 1

Division Algorithm For General Divisors

The division algorithm for general divisors is the same as that of the polynomial division algorithm discussed in the section of the division of one polynomial by another polynomial. One important fact about this division is that the degree of the divisor can be any positive integer lesser than the dividend.

Let us take an example: Let p(x) = x⁴ − 4x³ + 3x² + 2x − 1 be the dividend and g(x) = x² − 2x + 1 be the divisor. Here the degree of the divisor is 2, which is lesser than or equal to the dividend's degree. To know more about this division algorithm, please click here. We will divide p(x) by g(x) now.

Try verifying the division algorithm in this case.

Important Notes on Division Algorithm:

• A polynomial can be divided by another polynomial of a lower degree only.
• Arrange the dividend polynomial from the greatest to the lowest power before starting the division.
• If the divisor polynomial is not a factor of the dividend obtained at any step of the polynomial division then it means that a remainder other than 0 will be left behind.
• We can use the division algorithm to find one of the dividend, divisor, quotient, or remainder when the other three of these are given.

☛ Related Topics:

• Polynomial Division Calculator
• Synthetic Division
• Dividing Two Polynomials

What is the Division Algorithm Formula?

The division algorithm formula is: Dividend = (Divisor × Quotient) + Remainder. This can also be written as: p(x) = q(x) × g(x) + r(x), where,

• p(x) is the dividend.
• q(x) is the quotient.
• g(x) is the divisor.
• r(x) is the remainder.

How do You Verify a Division Algorithm?

To verify a division algorithm, we multiply the divisor to the quotient and add it to the remainder. This should result in the dividend.

Can Division Algorithm be Used for Polynomials?

Yes, polynomial division can also be verified using the division algorithm. Here, the degree of the dividend must be greater than or equal to that of the divisor.

How to Apply Division Algorithm?

Division algorithm says Dividend = (Divisor × Quotient) + Remainder. Hence, this can be used to find one of the following terms when the other 3 are given:

• Dividend
• Divisor
• Quotient
• Remainder

What are the Applications of the Division Algorithm?

The division algorithm can be used to find the HCF of two numbers in the easiest way. To learn the process of finding HCF using the division algorithm, click here.