Factor polynomial $$ 27r^{3}+216 $$

$$ 27r^{3}+216 = 27( r+2 )( r^{2}-2r+4 ) $$

Step 1 :

After factoring out $ 27 $ we have:

$$ 27r^{3}+216 = 27 ( r^{3}+8 ) $$

Step 2 :

To factor $ r^{3}+8 $ we can use sum of cubes formula:

$$ I^3 - II^3 = (I + II)(I^2 - I \cdot II + II^2) $$

After putting $ I = r $ and $ II = 2 $ , we have:

$$ r^{3}+8 = ( r+2 ) ( r^{2}-2r+4 ) $$

Step 3 :

Step 3: Identify constants $ \color{blue}{ b }$ and $\color{red}{ c }$. ( $ \color{blue}{ b }$ is a number in front of the $ x $ term and $ \color{red}{ c } $ is a constant). In our case:

$$ \color{blue}{ b = -2 } ~ \text{ and } ~ \color{red}{ c = 4 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -2 } $ and multiply to $ \color{red}{ 4 } $.

Step 4: Find out pairs of numbers with a product of $\color{red}{ c = 4 }$.

PRODUCT = 4
1     4	-1     -4
2     2	-2     -2

Step 5: Because none of these pairs will give us a sum of $ \color{blue}{ -2 }$, we conclude the polynomial cannot be factored.

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