SOLUTION: factor completely 150v^2+360vp+216p^2

Algebra.Com
Question 340411: factor completely
150v^2+360vp+216p^2
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!

Start with the given expression


Factor out the GCF


Now let's focus on the inner expression




------------------------------------------------------------



Looking at we can see that the first term is and the last term is where the coefficients are 25 and 36 respectively.

Now multiply the first coefficient 25 and the last coefficient 36 to get 900. Now what two numbers multiply to 900 and add to the middle coefficient 60? Let's list all of the factors of 900:



Factors of 900:
1,2,3,4,5,6,9,10,12,15,18,20,25,30,36,45,50,60,75,90,100,150,180,225,300,450

-1,-2,-3,-4,-5,-6,-9,-10,-12,-15,-18,-20,-25,-30,-36,-45,-50,-60,-75,-90,-100,-150,-180,-225,-300,-450 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 900
1*900
2*450
3*300
4*225
5*180
6*150
9*100
10*90
12*75
15*60
18*50
20*45
25*36
30*30
(-1)*(-900)
(-2)*(-450)
(-3)*(-300)
(-4)*(-225)
(-5)*(-180)
(-6)*(-150)
(-9)*(-100)
(-10)*(-90)
(-12)*(-75)
(-15)*(-60)
(-18)*(-50)
(-20)*(-45)
(-25)*(-36)
(-30)*(-30)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to 60? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 60

First NumberSecond NumberSum
19001+900=901
24502+450=452
33003+300=303
42254+225=229
51805+180=185
61506+150=156
91009+100=109
109010+90=100
127512+75=87
156015+60=75
185018+50=68
204520+45=65
253625+36=61
303030+30=60
-1-900-1+(-900)=-901
-2-450-2+(-450)=-452
-3-300-3+(-300)=-303
-4-225-4+(-225)=-229
-5-180-5+(-180)=-185
-6-150-6+(-150)=-156
-9-100-9+(-100)=-109
-10-90-10+(-90)=-100
-12-75-12+(-75)=-87
-15-60-15+(-60)=-75
-18-50-18+(-50)=-68
-20-45-20+(-45)=-65
-25-36-25+(-36)=-61
-30-30-30+(-30)=-60



From this list we can see that 30 and 30 add up to 60 and multiply to 900


Now looking at the expression , replace with (notice adds up to . So it is equivalent to )




Now let's factor by grouping:


Group like terms


Factor out the GCF of out of the first group. Factor out the GCF of out of the second group


Since we have a common term of , we can combine like terms

So factors to


So this also means that factors to (since is equivalent to )


note: is equivalent to since the term occurs twice. So also factors to



------------------------------------------------------------




So our expression goes from and factors further to


------------------
Answer:

So factors to

If you need more help, email me at jim_thompson5910@hotmail.com

Also, feel free to check out my tutoring website

Jim