LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

ALGEBRAIC - HIGHEST COMMON FACTOR (HCF)

**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.)**