LOWEST COMMON MULTIPLE (LCM) –
The LCM of the two or more algebraic expressions is the lowest expression which is exactly divisible by all the given expressions.
LCM of the Monomial –
To find the LCM of given monomials, the following steps are given below –
Step.1) Find the LCM of the numerical coefficients of the given monomials.
Step.2) Take the highest powers of each of the variables in the monomials.
Step.3) The product of the number and the powers of the variables obtained in Steps.1 & Steps.2 is the required LCM.
There are some example are given below for better understanding –
Example.1) Find the LCM of 2ab² & 3a²
Ans.) The LCM of monomials that have no factor in common is the product of the monomials.
The LCM of monomials coefficients 2 & 3 is = 2 X 3 = 6
The highest power of variable a = a²
The highest power of the variable b = b²
So, the desired LCM would be = 6 X a² X b² = 6a²b² (Ans.)
Example.2) Find the LCM of 3a⁴b⁵, 6aᶟc⁴, 9b²cᶟ
Ans.) The LCM of the numerical coefficients is = 3 X 2 X 3 = 18, where 3 = 3 X 1, 6 = 3 X 2 X 1, and 9 = 3 X 3 X 1.
The highest power of the variable a = a⁴
The highest power of variable b = b⁵
The highest power of variable c = c⁴
So, the desired LCM is = 18 X a⁴ X b⁵ X c⁴ = 18a⁴b⁵c⁴ (Ans.)
LCM of the Polynomial –
To find the LCM of given Polynomials, the following steps are given below –
Step.1) Find the LCM of the numerical coefficients (if any) of the polynomials.
Step.2) Factorize the given polynomials.
Step.3) Take the highest power of each of the factors (including the ones in common)
Step.4) The product of the number and the powers of the factors obtained in Step.1 & Step.3 is the LCM of the given polynomials.
There are some example are given below for your better understanding,
Example.1) Find the LCM of (2a² - a – 6), (3a² -7a + 2), and (6a²+ 7a -3)
Ans.) First of all we have to find the factors of given terms –
2a² - a – 6 = 2a² - (4 – 3)a – 6 = 2a² - 4a + 3a – 6
= 2a (a – 2) + 3 (a – 2)
= (a – 2) ( 2a + 3)
3a² -7a + 2 = 3a² - (6 + 1)a + 2
= 3a² - 6a – a + 2
= 3a (a – 2) – 1(a – 2)
= (a – 2) (3a – 1)
6a²+ 7a -3 = 6a² + (9 – 2)a – 3
= 6a² + 9a – 2a – 3
= 3a (2a + 3) – (2a + 3)
= (3a – 1) (2a + 3)
So, the LCM of the given expression = (a – 2) (3a – 1) (2a + 3) (Ans.)
Example.2) Find the LCM of (a²- 7a + 12), (3a² - 6a - 9), and (2aᶟ - 6a² - 8a)
First of all we have to find the factors of given terms –
a² - 7a + 12 = a² - (4 + 3)a + 12
= a² - 4a – 3a + 12
= a (a – 4) – 3 (a – 4) = (a – 3) (a – 4)
3a² - 6a – 9 = 3a² - (9 – 3)a – 9
= 3a² - 9x + 3a – 9
= 3a (a – 3) + 3 (a – 3) = 3 (a + 1) (a – 3)
2aᶟ - 6a² - 8a = 2a (a² - 3a – 4)
= 2a { a² - (4 – 1)a – 4}
= 2a { a² - 4a + a – 4}
= 2a {a(a – 4) + (a – 4)} = 2a (a + 1) (a – 4)
So, the desired LCM is = 2a (a + 1) (a – 3) (a – 4) (Ans.)