HCF of two algebraic expression

To find the Highest Common Factor i.e. H.C.F. of algebraic expressions, the

factorization method and process of continuous division are in practice in the
conventional system. We now apply' Lopana - Sthapana' Sutra, the 'Sankalana
vyavakalanakam' process and the 'Adyamadya' rule to find out the H.C.F in a
more easy and elegant way.


Question: Find the H.C.F. of x2 + 5x + 4 and x2 + 7x + 6.

1. Factorization method:


x2 + 5x + 4 = (x + 4) (x + 1)
x2 + 7x + 6 = (x + 6) (x + 1)

H.C.F. is ( x + 1 ).


2. Continuous division process.

x² + 5x + 4) x² + 7x + 6 ( 1

                     x²+ 5x + 4

                    ___________

                              2x + 2 ) x² + 5x + 4 ( ½x

                                            x²+   x

                                           __________

                                                     4x + 4) 2x + 2 ( ½

                                                                    2x + 2

                                                                   ______

                                                                         0

Thus 4x + 4 i.e., ( x + 1 ) is H.C.F.

3. Lopana - Sthapana process i.e. elimination and retention or alternate
destruction of the highest and the lowest powers is as below:

i.e.,, (x + 1) is H.C.F