Euclidean Algorithm for GCD of polynomials

elementary-number-theory gcd-and-lcm polynomials

Solution 1:

As $(a,b)=(a+n\cdot b,b),$ where $n$ is any integer

$$(2x^2+6x+3,2x+1=((2x+1)x+5x+3,2x+1)=(5x+3,2x+1)$$

Now, $2(5x+3)-5(2x+1)=1\implies(5x+3,2x+1)=1$

Solution 2:

You can observe that if you divide any polynomial then degree of remainder is less than that of divisor . If you divide $2x^2+6x+3$ by 2x+1 the remainder is 13/2 and they also don't have any common constants . So the GCD will be 1.

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