How do you find the LCM of k^2-2k-8, (k+2)^2?

Dec 19, 2017

$\textcolor{b l u e}{{\left(k + 2\right)}^{2} \cdot \left(k - 4\right)}$

Explanation:

We are given the following algebraic expressions

$\textcolor{red}{\begin{matrix}{k}^{2} - 2 k - 8 \\ {k + 2}^{2}\end{matrix}}$

We need to find the Least Common Multiple (LCM)

To find the LCM, we will use the following steps:

1. We must express each expression as the product of it's factors. If there are Prime Factors we must use them

2. We need to find the product of each factor with the highest power that occurs in the expressions

3. The product from above is our required LCM

Hence, LCM is the smallest expression that is divisible by each of the given algebraic expressions.

Let us first consider color(red)((k^2 - 2k -8)

We will break the above expression into appropriate groups as shown below:

$\left({k}^{2} + 2 k\right) + \left(- 4 k + 8\right)$

Factor Out as shown below:

$k \left(k + 2\right) - 4 \left(k + 2\right)$

Hence, we get the factors

$\textcolor{g r e e n}{\left(k + 2\right) \cdot \left(k - 4\right)}$... Result.1

Next, we will consider

$\textcolor{red}{{\left(k + 2\right)}^{2}}$

This can also be written as the product of two factors:

$\textcolor{g r e e n}{\left(k + 2\right) \cdot \left(k + 2\right)}$ ... Result.2

We will now compute an appropriate expression comprising of factors that appear either in ${\left(k + 2\right)}^{2} \mathmr{and} \left(k + 2\right) \left(k - 4\right)$

Hence, we get our LCM:

$\textcolor{b l u e}{{\left(k + 2\right)}^{2} \cdot \left(k - 4\right)}$

Hope this helps.