Factor polynomial $$ x^{3}-4x^{2}-21x $$

$$ x^{3}-4x^{2}-21x = x( x+3 )( x-7 ) $$

Step 1 :

After factoring out $ x $ we have:

$$ x^{3}-4x^{2}-21x = x ( x^{2}-4x-21 ) $$

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:

$$ \color{blue}{ b = -4 } ~ \text{ and } ~ \color{red}{ c = -21 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -4 } $ and multiply to $ \color{red}{ -21 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = -21 }$.

PRODUCT = -21
-1     21	1     -21
-3     7	3     -7

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

PRODUCT = -21 and SUM = -4
-1     21	1     -21
-3     7	3     -7

Step 5: Put 3 and -7 into placeholders to get factored form.

$$ \begin{aligned} x^{2}-4x-21 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}-4x-21 & = (x + 3)(x -7) \end{aligned} $$

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