Factor polynomial $$ 250m^{3}+54 $$

$$ 250m^{3}+54 = 2( 5m+3 )( 25m^{2}-15m+9 ) $$

Step 1 :

After factoring out $ 2 $ we have:

$$ 250m^{3}+54 = 2 ( 125m^{3}+27 ) $$

Step 2 :

To factor $ 125m^{3}+27 $ we can use sum of cubes formula:

$$ I^3 - II^3 = (I + II)(I^2 - I \cdot II + II^2) $$

After putting $ I = 5m $ and $ II = 3 $ , we have:

$$ 125m^{3}+27 = ( 5m+3 ) ( 25m^{2}-15m+9 ) $$

Step 3 :

Step 3: 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:

$$ a = 25 , b = -15 ~ \text{ and } ~ c = 9 $$

Step 4: Multiply the leading coefficient $\color{blue}{ a = 25 }$ by the constant term $\color{blue}{c = 9} $.

$$ a \cdot c = 225 $$

Step 5: Find out two numbers that multiply to $ a \cdot c = 225 $ and add to $ b = -15 $.

Step 6: All pairs of numbers with a product of $ 225 $ are:

PRODUCT = 225
1     225	-1     -225
3     75	-3     -75
5     45	-5     -45
9     25	-9     -25
15     15	-15     -15

Step 7: Find out which factor pair sums up to $\color{blue}{ b = -15 }$

Step 8: 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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