Factor polynomial $$ 3x^{3}-9x^{2}-54x $$

$$ 3x^{3}-9x^{2}-54x = 3x( x+3 )( x-6 ) $$

Step 1 :

After factoring out $ 3x $ we have:

$$ 3x^{3}-9x^{2}-54x = 3x ( x^{2}-3x-18 ) $$

Step 2 :

Step 2: 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:

$$ \color{blue}{ b = -3 } ~ \text{ and } ~ \color{red}{ c = -18 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -3 } $ and multiply to $ \color{red}{ -18 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = -18 }$.

PRODUCT = -18
-1     18	1     -18
-2     9	2     -9
-3     6	3     -6

Step 4: Find out which pair sums up to $\color{blue}{ b = -3 }$

PRODUCT = -18 and SUM = -3
-1     18	1     -18
-2     9	2     -9
-3     6	3     -6

Step 5: Put 3 and -6 into placeholders to get factored form.

$$ \begin{aligned} x^{2}-3x-18 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}-3x-18 & = (x + 3)(x -6) \end{aligned} $$

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