SOLUTION: factor completly 6z^3-27z^2+12z

Algebra ->  Equations -> SOLUTION: factor completly 6z^3-27z^2+12z      Log On


   

Question 110952: factor completly
6z^3-27z^2+12z
Found 2 solutions by jim_thompson5910, scott8148:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
6z%5E3-27z%5E2%2B12z Start with the given expression


3z%282z%5E2-9z%2B4%29 Factor out the GCF 3z

Now let's focus on 2z%5E2-9z%2B4

Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 2z%5E2-9z%2B4, we can see that the first coefficient is 2, the second coefficient is -9, and the last term is 4.



Now multiply the first coefficient 2 by the last term 4 to get %282%29%284%29=8.



Now the question is: what two whole numbers multiply to 8 (the previous product) and add to the second coefficient -9?



To find these two numbers, we need to list all of the factors of 8 (the previous product).



Factors of 8:

1,2,4,8

-1,-2,-4,-8



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to 8.

1*8 = 8
2*4 = 8
(-1)*(-8) = 8
(-2)*(-4) = 8


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -9:



First NumberSecond NumberSum
181+8=9
242+4=6
-1-8-1+(-8)=-9
-2-4-2+(-4)=-6




From the table, we can see that the two numbers -1 and -8 add to -9 (the middle coefficient).



So the two numbers -1 and -8 both multiply to 8 and add to -9



Now replace the middle term -9z with -z-8z. Remember, -1 and -8 add to -9. So this shows us that -z-8z=-9z.



2z%5E2%2Bhighlight%28-z-8z%29%2B4 Replace the second term -9z with -z-8z.



%282z%5E2-z%29%2B%28-8z%2B4%29 Group the terms into two pairs.



z%282z-1%29%2B%28-8z%2B4%29 Factor out the GCF z from the first group.



z%282z-1%29-4%282z-1%29 Factor out 4 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



%28z-4%29%282z-1%29 Combine like terms. Or factor out the common term 2z-1



===============================================================



Answer:



So 2%2Az%5E2-9%2Az%2B4 factors to %28z-4%29%282z-1%29.



In other words, 2%2Az%5E2-9%2Az%2B4=%28z-4%29%282z-1%29.



Note: you can check the answer by expanding %28z-4%29%282z-1%29 to get 2%2Az%5E2-9%2Az%2B4 or by graphing the original expression and the answer (the two graphs should be identical).




So 6z%5E3-27z%5E2%2B12z factors to 3z%28z-4%29%282z-1%29%29

Answer by scott8148(6628) About Me  (Show Source):
You can put this solution on YOUR website!
3z(2z-1)(z-4)