SOLUTION: Factor by grouping. Please help!Thank you(: 18n^2+57n-10!thanks again!

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Factor by grouping. Please help!Thank you(: 18n^2+57n-10!thanks again!      Log On


   

Question 299541: Factor by grouping. Please help!Thank you(:
18n^2+57n-10!thanks again!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 18n%5E2%2B57n-10, we can see that the first coefficient is 18, the second coefficient is 57, and the last term is -10.


Now multiply the first coefficient 18 by the last term -10 to get %2818%29%28-10%29=-180.


Now the question is: what two whole numbers multiply to -180 (the previous product) and add to the second coefficient 57?


To find these two numbers, we need to list all of the factors of -180 (the previous product).


Factors of -180:
1,2,3,4,5,6,9,10,12,15,18,20,30,36,45,60,90,180
-1,-2,-3,-4,-5,-6,-9,-10,-12,-15,-18,-20,-30,-36,-45,-60,-90,-180


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


These factors pair up and multiply to -180.
1*(-180) = -180
2*(-90) = -180
3*(-60) = -180
4*(-45) = -180
5*(-36) = -180
6*(-30) = -180
9*(-20) = -180
10*(-18) = -180
12*(-15) = -180
(-1)*(180) = -180
(-2)*(90) = -180
(-3)*(60) = -180
(-4)*(45) = -180
(-5)*(36) = -180
(-6)*(30) = -180
(-9)*(20) = -180
(-10)*(18) = -180
(-12)*(15) = -180

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 57:


First NumberSecond NumberSum
1-1801+(-180)=-179
2-902+(-90)=-88
3-603+(-60)=-57
4-454+(-45)=-41
5-365+(-36)=-31
6-306+(-30)=-24
9-209+(-20)=-11
10-1810+(-18)=-8
12-1512+(-15)=-3
-1180-1+180=179
-290-2+90=88
-360-3+60=57
-445-4+45=41
-536-5+36=31
-630-6+30=24
-920-9+20=11
-1018-10+18=8
-1215-12+15=3



From the table, we can see that the two numbers -3 and 60 add to 57 (the middle coefficient).


So the two numbers -3 and 60 both multiply to -180 and add to 57


Now replace the middle term 57n with -3n%2B60n. Remember, -3 and 60 add to 57. So this shows us that -3n%2B60n=57n.


18n%5E2%2Bhighlight%28-3n%2B60n%29-10 Replace the second term 57n with -3n%2B60n.


%2818n%5E2-3n%29%2B%2860n-10%29 Group the terms into two pairs.


3n%286n-1%29%2B%2860n-10%29 Factor out the GCF 3n from the first group.


3n%286n-1%29%2B10%286n-1%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.


%283n%2B10%29%286n-1%29 Combine like terms. Or factor out the common term 6n-1


===============================================================


Answer:


So 18n%5E2%2B57n-10 factors to %283n%2B10%29%286n-1%29.


In other words, 18n%5E2%2B57n-10=%283n%2B10%29%286n-1%29.


Note: you can check the answer by expanding %283n%2B10%29%286n-1%29 to get 18n%5E2%2B57n-10 or by graphing the original expression and the answer (the two graphs should be identical).