LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

ALGEBRAIC - LOWEST COMMON MULTIPLE (LCM)

__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.)**** **