Division Algorithm for Polynomials

Division Algorithm for Polynomials

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. 

Dividend is the product of the divisor and the quotient, added to the remainder (if any). The same rules apply for 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 the divisor. To verify the result, we multiply the divisor and the quotient and add it to the remainder, if any.

Arithmetic and polynomial division

Lesson Plan


What Is Division Algorithm of Polynomials?

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:

\(\begin{align}p(x) = q(x) \times g(x) + r(x)\end{align}\)

where, \(\begin{align}p(x)\end{align}\) is the dividend.

 \(\begin{align}q(x)\end{align}\) is the quotient.

 \(\begin{align}g(x)\end{align}\) is the divisor.

 \(\begin{align}r(x)\end{align}\) is the remainder.

If we compare this to the regular division of numbers, we can easily understand this as: Dividend = (Divisor X Quotient) + Remainder


Procedure To Divide a Polynomial by Another Polynomial

The steps for the polynomial division algorithm are given below.

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

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

Step 3: Find the second term of the quotient by dividing the largest degree term of the new dividend obtained in step 2 with the largest degree term of the divisor.

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

Let us understand this with an example.

Divide \(\begin{align}2x^3+3x^2+4x+3\end{align}\) by  \(\begin{align}x+1\end{align}\)

Here

\(\begin{align}p(x) = 2x^3+3x^2+4x+3\end{align}\)

\(\begin{align}g(x) = x+1\end{align}\)

Let us follow the division algorithm steps to perform the polynomial division.

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.

\(\begin{align}\therefore First \: term = \dfrac{2x^3}{x}\end{align}\)  \(\begin{align} = 2x^2\end{align}\)

Step 3: The new dividend is \(\begin{align}x^2+4x\end{align}\)

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.

\(\begin{align}\therefore Second \: term = \dfrac{x^2}{x}\end{align}\)  \(\begin{align} = x\end{align}\)

Step 5: The new dividend is  \(\begin{align}3x+3\end{align}\)

Step 6: The third term of the quotient is obtained by dividing the largest degree term of the new dividend obtained in step 4 with the largest degree term of the divisor.

\(\begin{align}\therefore Third \: term = \dfrac{3x}{x}\end{align}\)  \(\begin{align} =3\end{align}\)

Step 7: The quotient is \(\begin{align}2x^2+x+3\end{align}\)

Polynomial division example

\(\begin{align}p(x) = 2x^3+3x^2+4x+3\end{align}\) is the dividend

 \(\begin{align}q(x) = 2x^2+x+3\end{align}\) is the quotient

 \(\begin{align}g(x) = x+1\end{align}\) is the divisor

 \(\begin{align}r(x) = 0 \end{align}\) is the remainder

Divison Algorithm For Linear Divisors

When a polynomial of degree  \(\begin{align}n \geq 1 \end{align}\)  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 the fact that the divisor is of the degree 1.

Let us look at an example below.

Let

\(\begin{align}p(x) = x^2+ x + 1 \end{align}\) be the dividend

and  \(\begin{align}g(x) = x-1\end{align}\) be the divisor

Here the degree of the divisor is 1. Here  \(\begin{align}g(x) = x-1\end{align}\) is called a "Linear divisor".

To know more about this division algorithm, please click here.

Linear divisors division

Divison Algorithm For General Divisors

Division algorithm for general divisors is the same as that of the polynomial division alogorithm discussed under the section of 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

\(\begin{align}p(x) = x^4- 4x^3 + 3x^2+ 2x - 1 \end{align}\) be the dividend

and  \(\begin{align}g(x) = x^2- 2x + 1\end{align}\) be the divisor

Here the degree of the divisor is 2, which is lesser than the dividend's degree. 

To know more about this division algorithm, please click here.

General divisors division

 
important notes to remember
Important Notes
  1. A polynomial can be divided by another polynomial of a lower degree only.
  2. Arrange the dividend polynomial from the greatest to the lowest power before starting the division.
  3. 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.

Solved Examples

Example 1

 

 

Gabriel wants to divide the polynomial \(\begin{align}4x^3+18x^2+15x+28\end{align}\) by \(\begin{align}x+4\end{align}\). Can you help him?

Solution

Dividing polynomials example 1

\(\begin{align}\therefore\end{align}\) The quotient is \(\begin{align}4x^2+2x+7\end{align}\)
Example 2

 

 

In the following polynomial division \(\begin{align}\dfrac {x^4+2x^2+17x-48}{x+3}\end{align}\) we do not have the \(\begin{align}x^3\end{align}\) term. Is it possible to do the division?

Solution

Since we do not have the \(\begin{align}x^3\end{align}\) term let us make it as \(\begin{align}0x^3\end{align}\)

Polynomial example 2

\(\begin{align}\therefore\end{align}\) The quotient is \(\begin{align}x^3-3x^2+11x-16\end{align}\)
Example 3

 

 

Solve the following division using the division algorithm.

\(\begin{align}\dfrac {x^3-4x^2+6x-2}{x^2-2}\end{align}\)

Solution

Polynomial example 3

Quotient is \(\begin{align}x-4\end{align}\) 

Remainder is \(\begin{align}8x-10\end{align}\)

Example 4

 

 

For a polynomial division,

the divisor \(\begin{align}g(x) = 3x-2\end{align}\)

the quotient \(\begin{align}q(x) = 6x^2+4\end{align}\)

the remainder \(\begin{align}r(x) = 5 \end{align}\)

Find the dividend \(\begin{align} p(x) \end{align}\).

Solution

We know that,

\(\begin{align}p(x) = q(x) \times g(x) + r(x)\end{align}\)

Substituting the values we get,

\(\begin{align}p(x) = (6x^2+4) \times (3x-2)+ 5\end{align}\)

On multiplying \(\begin{align}q(x) \,\, and \,\, g(x)\end{align}\), we get

\(\begin{align}(18x^3+12x-12x^2-8)+5\end{align}\)

\(\begin{align}18x^3+12x-12x^2-8+5\end{align}\)

Therefore, the dividend is \(\begin{align}18x^3-12x^2+12x- 3\end{align}\)

The dividend is \(\begin{align}18x^3-12x^2+12x-3\end{align}\)
 
tips and tricks
Tips and Tricks
  1. To verify if the polynomial division is correct, multiply the divisor with the quotient and add it to the remainder and check if it results in the dividend.
  2. If the dividend does not contain a term next to the highest degree, prefix a 0 as the coefficient to that term. For example, the polynomial, 3x3- 4x +8 does not contain an xterm so while doing the division make the dividend as 3x3+ 0x2- 4x +8.

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 

 


Let's Summarize

The mini-lesson targeted the fascinating concept of division algorithm of polynomials. Division algorithm of polynomial takes the same form as arithmetic division except the dividends being polynomials.Trying out the solved examples and interactive questions would enrich your knowledge on the subject. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

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FAQs on Division Algorithm of Polynomials

1. What is the division algorithm formula?

The division algorithm formula is: Dividend = (Divisor X Quotient) + Remainder. This can also be written as: \(\begin{align}p(x) = q(x) \times g(x) + r(x)\end{align}\), where,

\(\begin{align}p(x)\end{align}\) is the dividend.

 \(\begin{align}q(x)\end{align}\) is the quotient.

 \(\begin{align}g(x)\end{align}\) is the divisor.

 \(\begin{align}r(x)\end{align}\) is the remainder.

2. 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.

3. How do you find the dividend when the divisor and the remainder are given?

To find the dividend, we multiply the divisor with the quotient and then add it to the remainder.

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