SOLUTION: Help Me Factor 5p2-p-18

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Question 547991: Help Me Factor 5p2-p-18
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 5p%5E2-p-18, we can see that the first coefficient is 5, the second coefficient is -1, and the last term is -18.


Now multiply the first coefficient 5 by the last term -18 to get %285%29%28-18%29=-90.


Now the question is: what two whole numbers multiply to -90 (the previous product) and add to the second coefficient -1?


To find these two numbers, we need to list all of the factors of -90 (the previous product).


Factors of -90:
1,2,3,5,6,9,10,15,18,30,45,90
-1,-2,-3,-5,-6,-9,-10,-15,-18,-30,-45,-90


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


These factors pair up and multiply to -90.
1*(-90) = -90
2*(-45) = -90
3*(-30) = -90
5*(-18) = -90
6*(-15) = -90
9*(-10) = -90
(-1)*(90) = -90
(-2)*(45) = -90
(-3)*(30) = -90
(-5)*(18) = -90
(-6)*(15) = -90
(-9)*(10) = -90

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -1:


First NumberSecond NumberSum
1-901+(-90)=-89
2-452+(-45)=-43
3-303+(-30)=-27
5-185+(-18)=-13
6-156+(-15)=-9
9-109+(-10)=-1
-190-1+90=89
-245-2+45=43
-330-3+30=27
-518-5+18=13
-615-6+15=9
-910-9+10=1



From the table, we can see that the two numbers 9 and -10 add to -1 (the middle coefficient).


So the two numbers 9 and -10 both multiply to -90 and add to -1


Now replace the middle term -1p with 9p-10p. Remember, 9 and -10 add to -1. So this shows us that 9p-10p=-1p.


5p%5E2%2Bhighlight%289p-10p%29-18 Replace the second term -1p with 9p-10p.


%285p%5E2%2B9p%29%2B%28-10p-18%29 Group the terms into two pairs.


p%285p%2B9%29%2B%28-10p-18%29 Factor out the GCF p from the first group.


p%285p%2B9%29-2%285p%2B9%29 Factor out 2 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-2%29%285p%2B9%29 Combine like terms. Or factor out the common term 5p%2B9


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Answer:


So 5p%5E2-p-18 factors to %28p-2%29%285p%2B9%29.


In other words, 5p%5E2-p-18=%28p-2%29%285p%2B9%29.


Note: you can check the answer by expanding %28p-2%29%285p%2B9%29 to get 5p%5E2-p-18 or by graphing the original expression and the answer (the two graphs should be identical).