Question

How do you find the LCM of\[{k^2} - 2k - 8\],\[{(k + 2)^2}\]?

Answer 1

Hint: Here in the above question we are asked to find the LCM of algebraic expressions. So firstly we will express each expression as products of its factor. So we will find the product of each factor with highest power occurring in the expressions. Then the required result that is LCM is the product obtained by using the definition of least common multiple.

Formula used : The formula for finding the LCM will be
\[(x,y) = \dfrac{{(x \times y)}}{{\gcd (x,y)}}\]

Complete step by step answer:
Since we are asked to find out the LCM of\[{k^2} - 2k - 8\],\[{(k + 2)^2}\]
Firstly we will consider \[({k^2} - 2k - 8)\]
Now after breaking the above equation into different group we get
\[({k^2} + 2k) + ( - 4k + 8)\]
Then we will factor out as shown here
\[k(k + 2) - 4(k + 2)\]
So the factors we will get will be
\[(k + 2)(k - 4)\]
The first required equation is \[(k + 2)(k - 4)\]
Later on we will consider \[{(k + 2)^2}\]
So again we will write it as product of two factors
\[(k + 2)(k + 2)\]
Lastly now we need to form an appropriate expression that comprise of factor appearing either in \[{(k + 2)^2}\]or \[(k + 2)(k - 4)\]
So the LCM will be \[{(k + 2)^2}(k - 4)\] is the required answer.

Additional Information:
LCM is the least common multiple and in arithmetic number theory it is the smallest integer \[k\] which is divisible by all the given numbers.

Note: We need to be careful while applying the method of factorization hence should use prime numbers for dividing. While solving such types of questions students make mistakes during multiplication. So avoid silly mistakes while breaking the equations into different groups and obtain the required result as asked in the question.