Factor polynomial $$ 12x^{5}-4x^{4}+28x^{3} $$

$$ 12x^{5}-4x^{4}+28x^{3} = 4x^{3}( 3x^{2}-x+7 ) $$

Step 1 :

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

$$ 12x^{5}-4x^{4}+28x^{3} = 4x^{3} ( 3x^{2}-x+7 ) $$

Step 2 :

Step 2: Identify constants $ a $ , $ b $ and $ c $.
$ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant. In this case:

$$ a = 3 , b = -1 ~ \text{ and } ~ c = 7 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 3 }$ by the constant term $\color{blue}{c = 7} $.

$$ a \cdot c = 21 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 21 $ and add to $ b = -1 $.

Step 5: All pairs of numbers with a product of $ 21 $ are:

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

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -1 }$

Step 7: Because none of these pairs will give us a sum of $ \color{blue}{ -1 }$, we conclude the polynomial cannot be factored.

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