Factor polynomial $$ 125x^{3}+75x^{2}+15x $$

$$ 125x^{3}+75x^{2}+15x = 5x( 25x^{2}+15x+3 ) $$

Step 1 :

After factoring out $ 5x $ we have:

$$ 125x^{3}+75x^{2}+15x = 5x ( 25x^{2}+15x+3 ) $$

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:

$$ a = 25 , b = 15 ~ \text{ and } ~ c = 3 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 25 }$ by the constant term $\color{blue}{c = 3} $.

$$ a \cdot c = 75 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 75 $ and add to $ b = 15 $.

Step 5: All pairs of numbers with a product of $ 75 $ are:

PRODUCT = 75
1     75	-1     -75
3     25	-3     -25
5     15	-5     -15

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 15 }$

Step 7: Because none of these pairs will give us a sum of $ \color{blue}{ 15 }$, we conclude the polynomial cannot be factored.

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