# GCD OF POLYNOMIALS USING DIVISION ALGORITHM

If f (x) and g(x) are two polynomials of same degree then the polynomial carrying the highest coefficient will be the dividend.

In case, if both have the same coefficient then compare the next least degree’s coefficient and proceed with the division.

If r (x) = 0 when f(x) is divided by g(x) then g(x) is called GCD of the polynomials.

Example 1 :

Find the GCD of the following polynomials.

x4 + 3x3 −x −3, x3 +x2 −5x + 3

Solution :

Let f(x)  =  x4 + 3x3 −x −3

g(x)  =  x3 +x2 −5x + 3

The degree of the polynomial f(x) is greater than g(x).

Note : If we have any missing term, we have to replace that place by 0.

The remainder is 3(x2 + 2x - 3), which is not equal to 0.So, we have to divide x3 +x2 −5x + 3 by x2 + 2x - 3 by leaving the remainder.

We get 0 as remainder by dividing the given polynomial by x2 + 2x - 3.

Hence the required GCD is x2 + 2x - 3.

Example 2 :

Find the GCD of the following polynomials.

x4  - 1,  x3 - 11x2 + x - 11

Solution :

Let f(x)  =  x4  - 1

g(x)  =  x3 - 11x2 + x - 11

The degree of the polynomial f(x) is greater than g(x).

The remainder is 120(x2 + 1), which is not equal to 0.So, we have to divide  x3 - 11x2 + x - 11 by 120(x2 + 1) by leaving the remainder.

Hence the G.C.D is x2 + 1

Example 3 :

Find the GCD of the following polynomials.

3x4 + 6x3 −12x2 −24x, 4x4 +14x3 + 8x2 −8x

Solution :

Let f(x)  =  3x4 + 6x3 −12x2 −24x

f(x)  =  3x(x3 + 2x2 - 4x - 8)

g(x)  =  4x4 +14x3 + 8x2 −8x

g(x)  =  2x (x3 + 7x2 + 4x - 4)

The remainder is 3(x2 + 4x + 4), which is not equal to 0.So, we have to divide  x3 + 2x2 - 4x - 8 by x2 + 4x + 4 by leaving the remainder.

By dividing x2 + 4x + 4, we get 0 remainder. The term x is also common for both the polynomials.

Hence the G.C.D is x(x2 + 4x + 4).

Example 4 :

Find the GCD of the following polynomials.

3x3 + 3x2 + 3x + 3 , 6x3 +12x2 + 6x +12

Solution :

Let f(x)  =  3x3 + 3x2 + 3x + 3

f(x)  =  3(x3 + x2 + x + 1)

g(x)  =  6x3 + 12x2 + 6x + 12

g(x)  =  6(x3 + 2x2 + x + 2)

The remainder is (x2 + 0x + 1), which is not equal to 0.So, we have to divide  x3 + x2 + x + 1 by x2 + 0x + 1 by leaving the remainder.

By dividing x2 + 1, we get 0 remainder. The constant 3 is also common for both the polynomials.

Hence the G.C.D is 3(x2 + 1).

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

## Recent Articles

1. ### First Fundamental Theorem of Calculus - Part 1

Apr 17, 24 11:27 PM

First Fundamental Theorem of Calculus - Part 1

2. ### Polar Form of a Complex Number

Apr 16, 24 09:28 AM

Polar Form of a Complex Number