SOLUTION: Here is the problem I don't understand. Factor this trinomial -45p2 - 12p + 12p3

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Question 243710: Here is the problem I don't understand.
Factor this trinomial
-45p2 - 12p + 12p3
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
-45p%5E2+-+12p+%2B+12p%5E3 Start with the given expression.


12p%5E3-45p%5E2+-+12p+ Rearrange the terms in descending degree.


3p%284p%5E2-15p-4%29 Factor out the GCF 3p.


Now let's try to factor the inner expression 4p%5E2-15p-4


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Looking at the expression 4p%5E2-15p-4, we can see that the first coefficient is 4, the second coefficient is -15, and the last term is -4.


Now multiply the first coefficient 4 by the last term -4 to get %284%29%28-4%29=-16.


Now the question is: what two whole numbers multiply to -16 (the previous product) and add to the second coefficient -15?


To find these two numbers, we need to list all of the factors of -16 (the previous product).


Factors of -16:
1,2,4,8,16
-1,-2,-4,-8,-16


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


These factors pair up and multiply to -16.
1*(-16) = -16
2*(-8) = -16
4*(-4) = -16
(-1)*(16) = -16
(-2)*(8) = -16
(-4)*(4) = -16

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -15:


First NumberSecond NumberSum
1-161+(-16)=-15
2-82+(-8)=-6
4-44+(-4)=0
-116-1+16=15
-28-2+8=6
-44-4+4=0



From the table, we can see that the two numbers 1 and -16 add to -15 (the middle coefficient).


So the two numbers 1 and -16 both multiply to -16 and add to -15


Now replace the middle term -15p with p-16p. Remember, 1 and -16 add to -15. So this shows us that p-16p=-15p.


4p%5E2%2Bhighlight%28p-16p%29-4 Replace the second term -15p with p-16p.


%284p%5E2%2Bp%29%2B%28-16p-4%29 Group the terms into two pairs.


p%284p%2B1%29%2B%28-16p-4%29 Factor out the GCF p from the first group.


p%284p%2B1%29-4%284p%2B1%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.


%28p-4%29%284p%2B1%29 Combine like terms. Or factor out the common term 4p%2B1


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So 3p%284p%5E2-15p-4%29 then factors further to 3p%28p-4%29%284p%2B1%29


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Answer:


So -45p%5E2+-+12p+%2B+12p%5E3 completely factors to 3p%28p-4%29%284p%2B1%29.


In other words, -45p%5E2+-+12p+%2B+12p%5E3=3p%28p-4%29%284p%2B1%29.


Note: you can check the answer by expanding 3p%28p-4%29%284p%2B1%29 to get -45p%5E2+-+12p+%2B+12p%5E3 or by graphing the original expression and the answer (the two graphs should be identical).