SOLUTION: I don't know how to factorise P^+5p-104

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Question 892354: I don't know how to factorise P^+5p-104
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
It can't be factorised as written
"P" is not the same as "p"
"P^" means nothing
If you mean
p^2+5p-104 then
You need to find factors of -104 which add up to 5
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression p%5E2%2B5p-104, we can see that the first coefficient is 1, the second coefficient is 5, and the last term is -104.



Now multiply the first coefficient 1 by the last term -104 to get %281%29%28-104%29=-104.



Now the question is: what two whole numbers multiply to -104 (the previous product) and add to the second coefficient 5?



To find these two numbers, we need to list all of the factors of -104 (the previous product).



Factors of -104:

1,2,4,8,13,26,52,104

-1,-2,-4,-8,-13,-26,-52,-104



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



These factors pair up and multiply to -104.

1*(-104) = -104
2*(-52) = -104
4*(-26) = -104
8*(-13) = -104
(-1)*(104) = -104
(-2)*(52) = -104
(-4)*(26) = -104
(-8)*(13) = -104


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 5:



First NumberSecond NumberSum
1-1041+(-104)=-103
2-522+(-52)=-50
4-264+(-26)=-22
8-138+(-13)=-5
-1104-1+104=103
-252-2+52=50
-426-4+26=22
-813-8+13=5




From the table, we can see that the two numbers -8 and 13 add to 5 (the middle coefficient).



So the two numbers -8 and 13 both multiply to -104 and add to 5



Now replace the middle term 5p with -8p%2B13p. Remember, -8 and 13 add to 5. So this shows us that -8p%2B13p=5p.



p%5E2%2Bhighlight%28-8p%2B13p%29-104 Replace the second term 5p with -8p%2B13p.



%28p%5E2-8p%29%2B%2813p-104%29 Group the terms into two pairs.



p%28p-8%29%2B%2813p-104%29 Factor out the GCF p from the first group.



p%28p-8%29%2B13%28p-8%29 Factor out 13 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%2B13%29%28p-8%29 Combine like terms. Or factor out the common term p-8



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Answer:



So p%5E2%2B5%2Ap-104 factors to %28p%2B13%29%28p-8%29.



In other words, p%5E2%2B5%2Ap-104=%28p%2B13%29%28p-8%29.



Note: you can check the answer by expanding %28p%2B13%29%28p-8%29 to get p%5E2%2B5%2Ap-104 or by graphing the original expression and the answer (the two graphs should be identical).