Euclid's Division Algorithm:Algebra form

Euclid's Division Algorithm

is also have it's utility in algebra in same formation.

whenever, one intended to perform long Division of algebraic terms then that procedure also follow Euclid Division Algorithm, or I can give you an idea to clear the concept so have a look at this,

when we divide 23 by 5
then this divided as
23/5 = 4 ³/5

here quotient is 4 and remainder is 3.
But to justify this division we tend to multiply quotient to Divisor and result is added to remainder hence
5×4+ 3 =23

this is what can be expressed in generalisation as

B×q+ r = A

What Euclid has given.

On the other aspect of algebra this can be applied similarly likewise.

But the representation is quiet different as each term is considered as polynomial in algebra and represented by P(x) that's why
p(x) is considered as Dividend
q(x) as quotient
r(x) as remainder
and
g(x) as Divisor

so the algorithm will be

p(x) = q(x) × g(x) + r(x)

this is for justification of long Division of algebraic terms.
Alternatively this formula is useful to evaluate any of the polynomial be it quotient​, be it remainder or Divisor​ if all the other three polynomials are given.
Suppose, we've to evaluate q(x), then

q(x) =  [ p(x) - r(x) ]/ g(x).
Now, over to you let me know by comment if I'm missing something.