Highest Common Factor (HCF) –
If certain factors are common to two or more expressions then they are called common factors of those expressions. The product of all such common factors is called the highest common factor or HCF of the given expression.
HCF of the Monomial –
To find HCF of the monomials, we will follow such steps which are given below –
Step.1) Find the HCF of the numerical coefficients of the monomials
Step.2) Find the highest power of each of the variables common to the monomials.
Step.3) The product of the number and the powers of the variables obtained in Step.1 and Step.2 is the required HCF
There are some example are given below for your better understanding –
Example.1) Find the HCF of 10abᶟ and 25 a²b⁴
Ans.) 10abᶟ = 2 X 5 X a X b X b X b, 25a²b⁴ = 5 X 5 X a X a X b X b X b X b
The HCF of the numerical coefficients of the monomials = HCF of 10 & 25 is = 5
The highest power of the variable a common to both = a
The highest power of the variable b common to both = b X b X b = bᶟ
The HCF of the 10abᶟ and 25 a²b⁴ = 5 X a X b X b X b = 5abᶟ (Ans.)
Example.2) Find the HCF of 3a²b⁴, 6bᶟ, 9a²b² & - 12a⁴
Ans.) 3a²b⁴ = 3 X a X a X b X b X b X b, 6bᶟ = 2 X 3 X b X b X b, 9a²b² = 3 X 3 X a X a X b X b,
-12a⁴ = (-) X 3 X 2 X 2 X a X a X a X a
The HCF of 3, 6, 9 & -12 is = 3
The highest power of a common to the monomials = a⁰ (where, there is no a in 6bᶟ)
The highest power of b common to the monomials = b⁰ (where, there is no b in -12a⁴)
So, the HCF of 3a²b⁴, 6bᶟ, 9a²b² & - 12a⁴ = 3 X a⁰ X b⁰
= 3 X 1 X 1 = 3 (Ans.)
HCF of the Polynomial –
We should follow the following steps which are given below to find HCF –
Step.1) Find the HCF of the common numerical coefficients if any,
Step.2) Factorize each of the given expressions and take the factors common to all of them.
Step.3) The product of the number and the factors obtained in Step.1 & Step.2 is the required HCF.
There are some examples are given below for your better understanding –
Example.1) Find HCF of 15 (ab - b²), 10 (a²-b²) & 25 (a²b - ab²)
Ans.) The HCF of the numerical coefficients of the given expressions –
= the HCF of 15, 10, & 25 is = 5
Also, the factors of the following terms are –
(ab - b²) = b X (a – b)
(a²-b²) = (a + b)(a – b)
(a²b - ab²) = a X b X (a – b)
So, we can observe that the common factor of the given terms or expressions is (a – b)
So, the HCF of the given expression = 5 (a – b) (Ans.)
Example.2) Find the HCF of 4(a² - b²), 8(a²- b²+ ac – bc) and 16(aᶟ - a²b + ab²- bᶟ)
Ans.) The HCF of the numerical coefficient of the given expressions
= HCF of 4, 8, & 16 is = 4
The factors of the given terms are –
a²- b² = (a + b) (a – b)
a²- b²+ ac – bc = (a + b) (a – b ) + (a – b ) c = (a – b) ( a + b + c )
aᶟ - a²b + ab² - bᶟ = a² (a – b) + b² (a – b) = (a – b) (a² + b²)
so, the common factor of the given terms or expression is (a – b)
so, HCF of the given terms are = 4 (a – b) (Ans.)
Example.3) Find the HCF of the a⁴ + 2a² + 1, a⁶ + a⁴ - a² - 1, and a⁴ - 1
Ans.) Here, the HCF of the numerical coefficient of the given expressions is = 1
The factors of the given terms are –
a⁴ + 2a² + 1 = (a²)² + 2 X a² X 1 + (1)²
[ applying the formula a² + 2ab + b² = (a+b)² ]
= (a² + 1)² = (a² + 1) (a² + 1)
a⁶ + a⁴ - a²- 1 = a⁴ (a²+ 1) – (a²+ 1) = (a²+ 1) (a⁴ - 1)
a⁴ - 1 = (a²)² - (1)² = (a² + 1) (a² - 1)
so, from the above expression we found that the common factor is (a²+ 1)
so, the HCF of the given expression is = 1 X (a²+ 1) = (a²+ 1) (Ans.)