Factor polynomial $$ -3x^{3}+39x^{2}-126x $$

$$ -3x^{3}+39x^{2}-126x = ( -3 )x( x-6 )( x-7 ) $$

Step 1 :

After factoring out $ -3x $ we have:

$$ -3x^{3}+39x^{2}-126x = -3x ( x^{2}-13x+42 ) $$

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:

$$ \color{blue}{ b = -13 } ~ \text{ and } ~ \color{red}{ c = 42 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -13 } $ and multiply to $ \color{red}{ 42 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = 42 }$.

PRODUCT = 42
1     42	-1     -42
2     21	-2     -21
3     14	-3     -14
6     7	-6     -7

Step 4: Find out which pair sums up to $\color{blue}{ b = -13 }$

PRODUCT = 42 and SUM = -13
1     42	-1     -42
2     21	-2     -21
3     14	-3     -14
6     7	-6     -7

Step 5: Put -6 and -7 into placeholders to get factored form.

$$ \begin{aligned} x^{2}-13x+42 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}-13x+42 & = (x -6)(x -7) \end{aligned} $$

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