Highest Common Factor of Monomials by Factorization

How to find the highest common factor of monomials by factorization?

Let us follow the following examples to know how to find the highest common factor (H.C.F.) or greatest common factor (G.C.F.) of monomials by factorization.


Solved examples of H.C.F. or G.C.F. of monomials by factorization:

1. Find the H.C.F. of the monomials 2ab and 6a2b2.

Solution:

2ab = 2 × a × b

6a2b2 = 2 × 3 × a × a × b × b

From the resolved factors of the above two monomials, the common factors are indicated by red color.

The common factors among two monomials are 2, a, b.

Therefore, the required H.C.F. = 2 × a × b = 2ab


2. Find the H.C.F. of the monomials 8x2y, 12x3y2 and 20x2y2z.

Solution:

The H.C.F. of numerical coefficients = The H.C.F. of 8, 12 and 20.

Since, 8 = 2 × 2 × 2 = 23, 12 = 2 × 2 × 3 = 22 × 31 and 20 = 2 × 2 × 5 = 22 × 51

Therefore, the H.C.F. of 8, 12 and 20 is 4

Now, the variables x and y are present in all the quantities. Out of these the highest common power of x is 2 and the highest common power if y is 1.

Therefore, the required H.C.F. = 4x2y1 = 4x2y

The method by which the H.C.F. of the monomials are determined can be formulated as follows:

(i) The H.C.F. of the numerical coefficients are to be determined at first.

(ii) Then the variables are to be written beside the coefficient with their highest common power or greatest common power.


Note:

According to the well known definition of H.C.F. or G.C.F. each term should be divisible by it, but there should be no common factor in the quotients thus obtained.

The fact can be verified, for example 2 we can observe that;

8x2y/4x2y = 2 12x3y2/4x2y = 3xy

20x2y2z/4x2y = 5yz

Here, the quotients are 2, 3xy and 5yz which have no common factor between them.

Similarly after finding the highest common factor of monomials by factorization we can verify the above fact.








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