Knowing the fact that LCM of 2 or more numbers, is the smallest number which could e found, so that this one to be divided by each number from the given set of numbers, we have to factor each of the given numbers, into it's prime factors.
Let's factor c+3=1*(c+3).
Now, let's factor c^2+4c+3. Noticing that the expression c^2+4c+3 is a second degree polynomial, we'll write it's equation and find it's roots in order to write the polynomial as a product of linear factors.
c1 = [-4+sqrt(16-12)]/2
c1 = (-4+2)/2
c1 = -1
c2 = (-4-2)/2
c2 = -3
So, the polynomial c^2+4c+3, could be writtenas:
c^2+4c+3 = (c-(-1))(c-(-3)) = (c+1)(c+3)
So, the prime factors of c^2+4c+3 are (c+1) and (c+3).
It is obvious that the number c^2+4c+3 could be divided by (c+3) and the number (c+3) could be divided by itself.
So, the LCM of the numbers c^2+4c+3 and c+3 is c+3.
There are 2 expressions:
c^2+4c+3 and c+3.
The least common multiple is an expression which is least possible mulple(LCM) of both c+3 and c^2+4x+3.
Since first expression is of higher degree we would like to see whether it has any factor, whether there is a highest common factor between the two expressions.
c^2+4c+3 = c+3c+c+3 = c(c+3)+1(c+3) = (c+3)(c+1)
So c+3 is therefore the HCF(hoghest common factor).
Therefore the LCM of the two expressions or numbers = product of the given two expresions or numbers/HCF of these two given expressions or numbers)
= (c^2+4c+3)(c+3)/(c+3) = c^2+4c+3 is, therefore, the LCM of c^2+4c+3 and c+3.
To find the least common multiples of the two given expression we first factorize the first expression in the following steps>
c^2 + 4c + 3
c^2 + c + 3c + 3
c(c + 1) + 3(c + 1)
(c + 1)(c + 3)
From above factors we see that one of the factors of the first expression is (c + 3), which is same as the second given expression.
Thus first expression is a multiple of second expression. Also the first expression is the least multiple of it self. Thus the first expression is the least common multiple of both the expression.
Least common multiple is: c^2 + 4c + 3