SOLUTION: Completely factor the polynomial 2z^3+14z^2-60z

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Question 112193: Completely factor the polynomial 2z^3+14z^2-60z
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
2z%5E3%2B14z%5E2-60z Start with the given expression


2z%28z%5E2%2B7z-30%29 Factor out the GCF 2z


Now let's focus on the inner expression z%5E2%2B7z-30


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


Looking at the expression z%5E2%2B7z-30, we can see that the first coefficient is 1, the second coefficient is 7, and the last term is -30.



Now multiply the first coefficient 1 by the last term -30 to get %281%29%28-30%29=-30.



Now the question is: what two whole numbers multiply to -30 (the previous product) and add to the second coefficient 7?



To find these two numbers, we need to list all of the factors of -30 (the previous product).



Factors of -30:

1,2,3,5,6,10,15,30

-1,-2,-3,-5,-6,-10,-15,-30



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



These factors pair up and multiply to -30.

1*(-30) = -30
2*(-15) = -30
3*(-10) = -30
5*(-6) = -30
(-1)*(30) = -30
(-2)*(15) = -30
(-3)*(10) = -30
(-5)*(6) = -30


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 7:



First NumberSecond NumberSum
1-301+(-30)=-29
2-152+(-15)=-13
3-103+(-10)=-7
5-65+(-6)=-1
-130-1+30=29
-215-2+15=13
-310-3+10=7
-56-5+6=1




From the table, we can see that the two numbers -3 and 10 add to 7 (the middle coefficient).



So the two numbers -3 and 10 both multiply to -30 and add to 7



Now replace the middle term 7z with -3z%2B10z. Remember, -3 and 10 add to 7. So this shows us that -3z%2B10z=7z.



z%5E2%2Bhighlight%28-3z%2B10z%29-30 Replace the second term 7z with -3z%2B10z.



%28z%5E2-3z%29%2B%2810z-30%29 Group the terms into two pairs.



z%28z-3%29%2B%2810z-30%29 Factor out the GCF z from the first group.



z%28z-3%29%2B10%28z-3%29 Factor out 10 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%2B10%29%28z-3%29 Combine like terms. Or factor out the common term z-3



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Answer:



So z%5E2%2B7%2Az-30 factors to %28z%2B10%29%28z-3%29.



In other words, z%5E2%2B7%2Az-30=%28z%2B10%29%28z-3%29.



Note: you can check the answer by expanding %28z%2B10%29%28z-3%29 to get z%5E2%2B7%2Az-30 or by graphing the original expression and the answer (the two graphs should be identical).





2z%28z%2B10%29%28z-3%29 Now replace z%5E2%2B7z-30 with %28z%2B10%29%28z-3%29


So 2z%5E3%2B14z%5E2-60z factors to 2z%28z%2B10%29%28z-3%29