Factor polynomial $$ 27z^{2}-21z+2 $$

$$ 27z^{2}-21z+2 = ( 3z-2 )( 9z-1 ) $$

Step 1: 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:

$$ a = 27 , b = -21 ~ \text{ and } ~ c = 2 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 27 }$ by the constant term $\color{blue}{c = 2} $.

$$ a \cdot c = 54 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 54 $ and add to $ b = -21 $.

Step 4: All pairs of numbers with a product of $ 54 $ are:

PRODUCT = 54
1     54	-1     -54
2     27	-2     -27
3     18	-3     -18
6     9	-6     -9

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -21 }$

PRODUCT = 54 and SUM = -21
1     54	-1     -54
2     27	-2     -27
3     18	-3     -18
6     9	-6     -9

Step 6: Replace middle term $ -21 x $ with $ -3x-18x $:

$$ 27x^{2}-21x+2 = 27x^{2}-3x-18x+2 $$

Step 7: Apply factoring by grouping. Factor $ 3x $ out of the first two terms and $ -2 $ out of the last two terms.

$$ 27x^{2}-3x-18x+2 = 3x\left(9x-1\right) -2\left(9x-1\right) = \left(3x-2\right) \left(9x-1\right) $$

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