Theory:

We have learnt how to find the GCD(HCF) of second and third degree expressions by the method of factorization. In this session, we shall learn how to find the GCD of a polynomial by the method of long division.
Let \(f(x)\) and \(g(x)\) be two polynomial such that \(deg(f(x)) \geq deg(g(x))\). Then, the divisor is \(g(x)\).
 
The steps to find the Greatest Common Divisor(GCD) or the Highest Common Factor(HCF) of two polynomials \(f(x)\) and \(g(x)\) is given by:
 
Step 1: Divide \(f(x)\) by \(g(x)\) to obtain \(f(x) = g(x)q(x) + r(x)\) where \(q(x)\) is the quotient and \(r(x)\) is the remainder. Then, \(deg[r(x)] < deg[g(x)]\). If the remainder \(r(x)\) is zero, then \(g(x)\) is the GCD of \(f(x)\) and \(g(x)\).
 
Step 2: If the remainder \(r(x)\) is a non - zero, divide \(g(x)\) by \(r(x)\) such that \(g(x) = r(x)q(x) + r_1(x)\) where \(r_1(x)\) is the new remainder. Then, \(deg[r_1(x)] < deg[r(x)]\). If the remainder \(r_1(x)\) is zero, then \(r(x)\) is the GCD.
 
Step 3: If \(r_1(x)\) is non - zero, repeat step \(2\) until we get the remainder zero.
Important!
The GCD of polynomials \(f(x)\) and \(g(x)\) can be written as \(GCD(f(x),g(x))\).
Example:
Find the GCD of the polynomials \(x^4 + 3x^3 - x - 3\) and \(x^3 + x^2 - 5x + 3\).
 
Solution:
 
Let \(f(x) = x^4 + 3x^3 − x − 3\) and \(g(x) = x^3 + x^2 − 5x + 3\).
 
Here, \(deg(f(x)) > deg(g(x))\). Then, the divisor is \(g(x)\).
 
Step 1: Divide \(f(x)\) by \(g(x)\).
 
x+2x3+x25x+3x4+3x3+0x2x3x4+x35x2+3x¯2x3+5x24x32x3+2x210x+6¯3x2+6x9x2+2x3
 
This implies that remainder \(x^2 + 2x - 3 \neq 0\).
 
Step 2: Since remainder is non-zero, then divide \(g(x)\) by \(r(x)\).
 
x1x2+2x3x3+x25x+3x3+2x23x¯x22x+3x22x+3¯0
 
Since the remainder is zero, the gcd is \(x^2 + 2x - 3\).
 
Therefore, the \(GCD(f(x), g(x)) = x^2 + 2x - 3\).