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 6a^{2}b^{2}.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
Solution:
The H.C.F. of numerical coefficients = The H.C.F. of 8, 12 and 20.
Since, 8 = 2 × 2 × 2 = 2^{3}, 12 = 2 × 2 × 3 = 2^{2} × 3^{1} and 20 = 2 × 2 × 5 = 2^{2} × 5^{1}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. = 4x^{2}y^{1} = 4x^{2}yThe 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;
8x^{2}y/4x^{2}y = 2 12x^{3}y^{2}/4x^{2}y = 3xyHere, 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.
`8th Grade Math Practice
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