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

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: factor completely 150v^2+360vp+216p^2       Log On


   

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

150v%5E2%2B360vp%2B216p%5E2 Start with the given expression


6%2825v%5E2%2B60vp%2B36p%5E2%29 Factor out the GCF 6


Now let's focus on the inner expression 25v%5E2%2B60vp%2B36p%5E2




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



Looking at 25v%5E2%2B60vp%2B36p%5E2 we can see that the first term is 25v%5E2 and the last term is 36p%5E2 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 25v%5E2%2B60vp%2B36p%5E2, replace 60vp with 30vp%2B30vp (notice 30vp%2B30vp adds up to 60vp. So it is equivalent to 60vp)

25v%5E2%2Bhighlight%2830vp%2B30vp%29%2B36p%5E2


Now let's factor 25v%5E2%2B30vp%2B30vp%2B36p%5E2 by grouping:


%2825v%5E2%2B30vp%29%2B%2830vp%2B36p%5E2%29 Group like terms


5v%285v%2B6p%29%2B6p%285v%2B6p%29 Factor out the GCF of 5v out of the first group. Factor out the GCF of 6p out of the second group


%285v%2B6p%29%285v%2B6p%29 Since we have a common term of 5v%2B6p, we can combine like terms

So 25v%5E2%2B30vp%2B30vp%2B36p%5E2 factors to %285v%2B6p%29%285v%2B6p%29


So this also means that 25v%5E2%2B60vp%2B36p%5E2 factors to %285v%2B6p%29%285v%2B6p%29 (since 25v%5E2%2B60vp%2B36p%5E2 is equivalent to 25v%5E2%2B30vp%2B30vp%2B36p%5E2)


note: %285v%2B6p%29%285v%2B6p%29 is equivalent to %285v%2B6p%29%5E2 since the term 5v%2B6p occurs twice. So 25v%5E2%2B60vp%2B36p%5E2 also factors to %285v%2B6p%29%5E2



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




So our expression goes from 6%2825v%5E2%2B60vp%2B36p%5E2%29 and factors further to 6%285v%2B6p%29%5E2


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

So 150v%5E2%2B360vp%2B216p%5E2 factors to 6%285v%2B6p%29%5E2

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

Also, feel free to check out my tutoring website

Jim