In this chapter, we will learn to find greatest common factor for given set of monomial using solved examples.
To understand the chapter fully, you should have basic knowledge about the concept of HCF and monomials.
Greatest common monomial factor
In this topic, we will try to find the common factors present in the given monomials and then combine them.
To find the greatest common factor, follow the below steps;
(a) First find HCF of coefficients.
Separate all the coefficient from given monomial and then find the HCF.
The HCF tells the highest factor present in the given coefficient.
(b) Find common variable terms.
Select the particular variable and compare its exponents present in all monomials.
Select the variable will lowest power as it ensures that the factor is common among all the monomials.
Now combine the calculated coefficients and variables for final result.
I hope you got the basic idea of the process. Let us see some examples for further clarity.
GCF of Monomial – Solved examples
Example 01
Find greatest common factor of following monomials.
\mathtt{( a) \ \ 6x^{2} y}\\\ \\ \mathtt{( b) \ \ 2x^{3} y^{3}}\\\ \\ \mathtt{( c) \ \ 3x^{2} y^{2}}
Solution
Given above are three monomials. To find the GCF, follow the below steps;
(a) Find HCF of coefficient.
HCF (6, 2, 3) = 1
Hence, 1 is the highest number which is present in all the three monomial.
(b) Now find GCF of variables.
To find the GCF, select the variable with lowest coefficient.
Variable x
Among given monomials, the lowest exponent is 2.
This means that \mathtt{x^{2}} is present in all the three monomials.
Variable y
Among given monomial, the lowest exponent is 1.
Combining all the components we get;
\mathtt{\Longrightarrow \ 1.x^{2} .y}\\\ \\ \mathtt{\Longrightarrow \ x^{2} y}
Hence, \mathtt{x^{2} y} is the highest common factor among given variables.
Example 02
Find the greatest common factors of given monomials.
\mathtt{a) \ \ 8m^{5} n^{4}}\\\ \\ \mathtt{( b) \ \ 16\ m^{8} n^{5}}\\\ \\ \mathtt{( c) \ \ 4\ m^{9} n^{6}}
Solution
To find the greatest common factor, follow the below steps;
(a) Find HCF of coefficient.
HCF (8, 16, 4) = 4
Hence, number 4 is the common factor among given monomials.
(b) Find highest common variable factors
Here you have to choose variable with the lowest power.
Variable m
Lowest power is 5
Variable n
Lowest power is 4
Combining all the components, we get;
\mathtt{\Longrightarrow \ 4.m^{5} .n^{4}}\\\ \\ \mathtt{\Longrightarrow \ 4.m^{5} .n^{4}}
Hence, \mathtt{\Longrightarrow \ 4.m^{5} .n^{4}} is the highest common factor among given monomials.
Example 03
Find GCF of below monomials
\mathtt{( a) \ \ 15\ x^{2} y^{8} z^{3}}\\\ \\ \mathtt{( b) \ -27\ xy^{2} z}
Solution
Follow the below steps;
(a) Find HCF of coefficient
HCF (-27, 15) = 3
(b) Find highest common variable factors.
Choose the variables with lowest power.
Variable x
Lowest power is 1
Variable y
Lowest power is 2
Variable z
Lowest power is 1
Combining all the components we get;
\mathtt{\Longrightarrow \ 3\ .x\ .\ y^{2} .\ z}\\\ \\ \mathtt{\Longrightarrow 3\ x\ y^{2} \ z\ }
Hence, \mathtt{3\ x\ y^{2} \ z} is the highest common factor.
Example 04
Find the highest common factor of below monomials
\mathtt{( a) \ \ 60\ x\ y}\\\ \\ \mathtt{( b) \ 12\ x^{2} y^{3}}\\\ \\ \mathtt{( c) \ \ -16\ x^{3}}
Solution
Follow the below step;
(a) Find HCF of coefficient
HCF (60, 12, -16 ) = 4
Hence, 4 is the greatest number present in all the monomial.
(b) Highest common factor among monomial.
Here you have to select the lowest possible exponent.
Variable x
Lowest given exponent is 1.
Variable y
Lowest exponent is 0
Combining all the exponents we get 4x as the highest common factor.
Example 05
Find the highest common factor among given monomial.
\mathtt{( a) \ \ -20\ x^{7} \ y^{6}}\\\ \\ \mathtt{( b) \ -12\ x^{8} y^{5}}
Solution
Follow the below steps;
(a) Find HCF of coefficient.
HCF (-20, -12) = -4
Hence, -4 is the highest common factor present in both monomials.
(b) GCF of variables.
Here you have to select the lowest exponent value.
Variable x
Lowest exponent is 7.
Variable y
Lowest exponent is 5.
Combining all the components we get;
\mathtt{\Longrightarrow \ 4.x^{7} .y^{5}}\\\ \\ \mathtt{\Longrightarrow \ 4x^{7} y^{5}}