Factor polynomial $$ 343x^{3}-27 $$

$$ 343x^{3}-27 = ( 7x-3 )( 49x^{2}+21x+9 ) $$

Step 1 :

To factor $ 343x^{3}-27 $ we can use difference of cubes formula:

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

After putting $ I = 7x $ and $ II = 3 $ , we have:

$$ 343x^{3}-27 = ( 7x-3 ) ( 49x^{2}+21x+9 ) $$

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:

$$ a = 49 , b = 21 ~ \text{ and } ~ c = 9 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 49 }$ by the constant term $\color{blue}{c = 9} $.

$$ a \cdot c = 441 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 441 $ and add to $ b = 21 $.

Step 5: All pairs of numbers with a product of $ 441 $ are:

PRODUCT = 441
1     441	-1     -441
3     147	-3     -147
7     63	-7     -63
9     49	-9     -49
21     21	-21     -21

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 21 }$

Step 7: Because none of these pairs will give us a sum of $ \color{blue}{ 21 }$, we conclude the polynomial cannot be factored.

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