Factor polynomial $$ 4a^{3}-a^{2}b-36a+9b $$

$$ 4a^{3}-a^{2}b-36a+9b = (a+3)(a-3)(4a-b) $$

Step 1 :

To factor $ 4a^3-a^2b-36a+9b $ we can use factoring by grouping.

Group $ \color{blue}{ 4a^3 }$ with $ \color{blue}{ -a^2b }$ and $ \color{red}{ -36a }$ with $ \color{red}{ 9b }$ then factor each group.

$$ \begin{aligned} 4a^3-a^2b-36a+9b &= ( \color{blue}{ 4a^3-a^2b } ) + ( \color{red}{ -36a+9b }) = \\ &= \color{blue}{ a^2( 4a-b )} \color{red}{ -9( 4a-b ) } = \\ &= (a^2-9)(4a-b) \end{aligned} $$

Step 2 :

Rewrite $ a^2-9 $ as:

$$ \color{blue}{ a^2-9 = (a)^2 - (3)^2 } $$

Now we can apply the difference of squares formula.

$$ I^2 - II^2 = (I - II)(I + II) $$

After putting $ I = a $ and $ II = 3 $ , we have:

$$ a^2-9 = (a)^2 - (3)^2 = ( a-3 ) ( a+3 ) $$

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