 # Division Algorithm for Polynomials – Definition, Facts, Formula | Division Algorithm Problems and Solutions

In this platform, you will learn about the division algorithm. The division is one of the four basic mathematical operations, the other three being addition, subtraction, and multiplication. In simple words, division can be defined as the splitting of a large group into smaller groups such that every group will have an equal number of items. It is an operation that is used for equal grouping and equal sharing in math.

Let us learn about the 10th Grade Math Concept division algorithm definition, steps to divide a polynomial by another polynomial, division algorithm for linear divisors, division algorithm for general divisors, some solved example problems, and so on in detail on this Page.

### What is Division Algorithm?

Division Algorithm is an equation that forms a relationship between all four parts of the division. The division algorithm states that dividends can be expressed as the sum of the remainder to the product of quotient and divisor. It will be represented in the formula as,
Dividend = (Divisor × Quotient) + Remainder.

Similarly, in the case of polynomials, the division algorithm is used to determine the quotient polynomial and the constant remainder. Let the p(x), q(x), g(x) and r(x) represent as the dividend polynomial, quotient polynomial, divisor polynomial and remainder respectively. Hence, the division algorithm for them is given as, p(x) = g(x) × q(x) + r(x). where,

• r(x) = 0 which means degree of r(x) < degree of g(x).
• p(x) represents the dividend polynomial
• g(x) represents the divisor polynomial
• q(x) represents the quotient polynomial
• r(x) represents the remainder polynomial

Consider an example and see if it satisfies the above division algorithm or not. Divide 17 by 3. 17 divided by 3 will give us 5 as the quotient and 2 as the remainder.
Solution: Dividend = (Divisor × Quotient) + Remainder
17 = (3 × 5) + 2
17 = 15 + 2
17 = 17. Hence it is verified.

### How to Divide a Polynomial by Another Polynomial?

The steps to divide a polynomial by another polynomial are:
Step 1: Initially arrange the terms of dividend and the divisor will be in the decreasing order of their degrees.
Step 2: Next, to obtain the first term of quotient divides the highest degree term of the dividend by the highest degree term of the divisor.
Step 3: To obtain the second term of the quotient, divide the highest degree term of the new dividend obtained as a remainder by the highest degree term of the divisor.
Step 4: Continue this process until the degree of remainder is less than the degree of divisor.

### Division Algorithm for Linear Divisors

When a polynomial degree n≥1 is divided by a divisor with degree 1, then we call it division by the 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 degree will be 1. Below are the steps to divide polynomials with linear divisors are:

• First, we arrange the terms of the dividend and divisor in descending order of their powers.
• In the first term of the quotient, a polynomial is determined by dividing the highest degree term of the dividend and the largest degree term of the divisor.
• The next subsequent term of the quotient is by dividing the highest term of the new dividend obtained from the above steps with the largest term of the divisor.
• These steps are repeated until we obtain the degree of remainder less than the degree of divisor.

Look at an example below,
Let the p(x) = x2+x+1 be the dividend and the g(x) is x−1 be the divisor.
Here the degree of the divisor is 1.
Here, g(x) = x−1 is called a “Linear divisor”.

### 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 on the division of one polynomial by another polynomial. The important fact about this division is, that the degree of the divisor can be any positive integer lesser but it will be than the dividend. Consider an example:

Let the p(x) =x4-4x3+3x2+2x−1 be the dividend, and
the g(x)=x2−2x+1be the divisor
In this, the degree of the divisor is 2, which is lesser than the dividend’s degree.

### Division Algorithm Examples

Problem 1: In a polynomial division, the divisor is g(x) = 3x-2, the quotient is q(x)=6x2+4, and the remainder is r(x)=5. Find the value of the dividend p(x).
Solution:
As given in the question,
g(x) = 3x-2
q(x)=6x2+4
r(x)=5
Now, we need to find out the value of the dividend which is p(x).
We know that, p(x) = q(x) × g(x) + r(x)
Substituting the values in the above formula. we get,
p(x)=(6x2+4)×(3x−2)+5
On multiplying q(x) and g(x), we get
(18x3+12x −12x2−8)+5
18x3+12x−12x2 − 8+5 i.e., 18x3+12x−12x2 − 3
Therefore, the value of the dividend is 18x3−12x2+12x−3.

Problem 2: Apply Division Algorithm to find the quotient of q(x) and the remainder is r(x) on dividing the p(x) by g(x) as given below:
p(x) = x3−3x2+4x+2, g(x) = x2+2x+1.

Solution:
In the given question,
p(x)=x3−3x2+4x+2,
g(x)=x2+2x+1
Now, we will find the quotient and the remainder.
According to the question, the Dividend =x3−3x2+4x+2 and the Divisor =x2+2x+1.
In this, the dividend and divisor both are in the standard form.
Now, divide p(x) by g(x) we get the quotient and remainder values.
So, the Quotient is x−5 and the Remainder is 13x+7.

Problem 3: Give examples of polynomials p(x), q(x) and r(x), which satisfy the division algorithm and
(i) deg p(x) = deg q(x)
(ii) deg q(x) = deg r(x)
(iii) deg q(x) = 0

Solution:
In the given question,
(i) Consider q(x) = 3x2 + 2x + 6, and the degree of the q(x) is 2.
Next, p(x) = 12x2 + 8x + 24, the degree of p(x) is 2.
So, here deg p(x) = deg q(x).

(ii) Next, p(x) = x5 + 2x4 + 3x3+ 5x2 + 2
The q(x) is x2 + x + 1, degree of q(x) = 2
The g(x) is x3 + x2 + x + 1
r(x) = 2x2 – 2x + 1, and the degree of r(x) is 2.
Here, deg q(x) = deg r(x). Because both the degrees are same.

(iii) Let p(x) = 2x4 + x3 + 6x2 + 4x + 12
The q(x) = 2, and the degree of q(x) = 0.
The g(x) = x4 + 4x3 + 3x2 + 2x + 6
r(x) = 0
Here, deg q(x) = 0. Because it has zero degree value.

Problem 4: Obtain all the zeroes of 3x4 + 6x3 – 2x2 – 10x – 5, if two of its zeroes are √5/3 and −√5/3.

Solution:
In the given question,
Since two zeroes are √5/3 and −√5/3
x = −√5/3, x = −√5/3
⇒(x−√5/3)(x +√5/3)=x2−5/3 or 3x2 – 5 is a factor of the given polynomial.
Now, we will apply the division algorithm to the given polynomial.
So, 3x4 + 6x3 – 2x2 – 10x – 5
= (3x2 – 5) (x2 + 2x + 1) + 0
The Quotient is x2 + 2x + 1 = (x + 1)2
Zeroes of (x + 1)2 are –1, –1.
Hence, all its zeroes are √5/3, −√5/3, –1, –1.

Problem 5: On dividing x3 – 3x2 + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and –2x + 4, respectively. Find g(x).

Solution:
As given in the question,
p(x) = x3 – 3x2 + x + 2 q(x) = x – 2 and r (x) = –2x + 4
By Division Algorithm, we know that
p(x) = q(x) × g(x) + r(x)
Therefore,
x3 – 3x2 + x + 2 = (x – 2) × g(x) + (–2x + 4)
⇒ x3 – 3x2 + x + 2 + 2x – 4 = (x – 2) × g(x)
⇒g(x)=x3−3x2+3x−2/x−2
On dividing x3 – 3x2 + x + 2 by x – 2,
we get g(x) value. After dividing the g(x) value is  x2 – x + 1.
Therefore, the given polynomial g(x) is x2 – x + 1.

### FAQs on Division Algorithm

1. What is Division Algorithm for Polynomials?
If p(x), q(x), g(x) and r(x) represent, dividend polynomial, quotient polynomial, divisor polynomial and remainder respectively, the division algorithm for them is given as, p(x) = g(x) × q(x) + r(x).

2. How Do You Divide Polynomials with Linear Divisors?
The steps to divide polynomials with linear divisors are:

• Step 1: First we arrange the terms of the dividend and divisor in descending order of their powers.
• Step 2: The first term of the quotient polynomial is determined by dividing the highest degree term of the dividend and the largest degree term of the divisor.
• Step 3: The next subsequent term of the quotient by dividing the highest term of the new dividend obtained from the above steps with the largest term of the divisor.

3. What are Linear Divisors?
The linear divisors are linear polynomials that are written in a general form ax+b having their degree as 1.

4. 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 a dividend.

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