SOLUTION: 3p^2+17p-56

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Question 778400: 3p^2+17p-56
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming you want to factor this.



Looking at the expression 3p%5E2%2B17p-56, we can see that the first coefficient is 3, the second coefficient is 17, and the last term is -56.


Now multiply the first coefficient 3 by the last term -56 to get %283%29%28-56%29=-168.


Now the question is: what two whole numbers multiply to -168 (the previous product) and add to the second coefficient 17?


To find these two numbers, we need to list all of the factors of -168 (the previous product).


Factors of -168:
1,2,3,4,6,7,8,12,14,21,24,28,42,56,84,168
-1,-2,-3,-4,-6,-7,-8,-12,-14,-21,-24,-28,-42,-56,-84,-168


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


These factors pair up and multiply to -168.
1*(-168) = -168
2*(-84) = -168
3*(-56) = -168
4*(-42) = -168
6*(-28) = -168
7*(-24) = -168
8*(-21) = -168
12*(-14) = -168
(-1)*(168) = -168
(-2)*(84) = -168
(-3)*(56) = -168
(-4)*(42) = -168
(-6)*(28) = -168
(-7)*(24) = -168
(-8)*(21) = -168
(-12)*(14) = -168

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 17:


First NumberSecond NumberSum
1-1681+(-168)=-167
2-842+(-84)=-82
3-563+(-56)=-53
4-424+(-42)=-38
6-286+(-28)=-22
7-247+(-24)=-17
8-218+(-21)=-13
12-1412+(-14)=-2
-1168-1+168=167
-284-2+84=82
-356-3+56=53
-442-4+42=38
-628-6+28=22
-724-7+24=17
-821-8+21=13
-1214-12+14=2



From the table, we can see that the two numbers -7 and 24 add to 17 (the middle coefficient).


So the two numbers -7 and 24 both multiply to -168 and add to 17


Now replace the middle term 17p with -7p%2B24p. Remember, -7 and 24 add to 17. So this shows us that -7p%2B24p=17p.


3p%5E2%2Bhighlight%28-7p%2B24p%29-56 Replace the second term 17p with -7p%2B24p.


%283p%5E2-7p%29%2B%2824p-56%29 Group the terms into two pairs.


p%283p-7%29%2B%2824p-56%29 Factor out the GCF p from the first group.


p%283p-7%29%2B8%283p-7%29 Factor out 8 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%2B8%29%283p-7%29 Combine like terms. Or factor out the common term 3p-7


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Answer:


So 3p%5E2%2B17p-56 factors to %28p%2B8%29%283p-7%29.


In other words, 3p%5E2%2B17p-56=%28p%2B8%29%283p-7%29.


Note: you can check the answer by expanding %28p%2B8%29%283p-7%29 to get 3p%5E2%2B17p-56 or by graphing the original expression and the answer (the two graphs should be identical).