Factor polynomial $$ 3x^{6}-33x^{5}+72x^{4} $$

$$ 3x^{6}-33x^{5}+72x^{4} = 3x^{4}( x-3 )( x-8 ) $$

Step 1 :

After factoring out $ 3x^{4} $ we have:

$$ 3x^{6}-33x^{5}+72x^{4} = 3x^{4} ( x^{2}-11x+24 ) $$

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:

$$ \color{blue}{ b = -11 } ~ \text{ and } ~ \color{red}{ c = 24 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -11 } $ and multiply to $ \color{red}{ 24 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = 24 }$.

PRODUCT = 24
1     24	-1     -24
2     12	-2     -12
3     8	-3     -8
4     6	-4     -6

Step 4: Find out which pair sums up to $\color{blue}{ b = -11 }$

PRODUCT = 24 and SUM = -11
1     24	-1     -24
2     12	-2     -12
3     8	-3     -8
4     6	-4     -6

Step 5: Put -3 and -8 into placeholders to get factored form.

$$ \begin{aligned} x^{2}-11x+24 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}-11x+24 & = (x -3)(x -8) \end{aligned} $$

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