Factor polynomial $$ 6z^{4}-10z^{3}+4z^{2} $$

$$ 6z^{4}-10z^{3}+4z^{2} = 2z^{2}( z-1 )( 3z-2 ) $$

Step 1 :

After factoring out $ 2z^{2} $ we have:

$$ 6z^{4}-10z^{3}+4z^{2} = 2z^{2} ( 3z^{2}-5z+2 ) $$

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:

$$ a = 3 , b = -5 ~ \text{ and } ~ c = 2 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 3 }$ by the constant term $\color{blue}{c = 2} $.

$$ a \cdot c = 6 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 6 $ and add to $ b = -5 $.

Step 5: All pairs of numbers with a product of $ 6 $ are:

PRODUCT = 6
1     6	-1     -6
2     3	-2     -3

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -5 }$

PRODUCT = 6 and SUM = -5
1     6	-1     -6
2     3	-2     -3

Step 7: Replace middle term $ -5 x $ with $ -2x-3x $:

$$ 3x^{2}-5x+2 = 3x^{2}-2x-3x+2 $$

Step 8: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ -1 $ out of the last two terms.

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

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