SOLUTION: Directions are to factor each polynomial. If the polynomial cannot be factored using integers, write prime. 36m^2+60mn+25n^2

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Question 143547: Directions are to factor each polynomial. If the polynomial cannot be factored using integers, write prime.
36m^2+60mn+25n^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at 36m%5E2%2B60mn%2B25n%5E2 we can see that the first term is 36m%5E2 and the last term is 25n%5E2 where the coefficients are 36 and 25 respectively.

Now multiply the first coefficient 36 and the last coefficient 25 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 36m%5E2%2B60mn%2B25n%5E2, replace 60mn with 30mn%2B30mn (notice 30mn%2B30mn adds up to 60mn. So it is equivalent to 60mn)

36m%5E2%2Bhighlight%2830mn%2B30mn%29%2B25n%5E2


Now let's factor 36m%5E2%2B30mn%2B30mn%2B25n%5E2 by grouping:


%2836m%5E2%2B30mn%29%2B%2830mn%2B25n%5E2%29 Group like terms


6m%286m%2B5n%29%2B5n%286m%2B5n%29 Factor out the GCF of 6m out of the first group. Factor out the GCF of 5n out of the second group


%286m%2B5n%29%286m%2B5n%29 Since we have a common term of 6m%2B5n, we can combine like terms

So 36m%5E2%2B30mn%2B30mn%2B25n%5E2 factors to %286m%2B5n%29%286m%2B5n%29


So this also means that 36m%5E2%2B60mn%2B25n%5E2 factors to %286m%2B5n%29%286m%2B5n%29 (since 36m%5E2%2B60mn%2B25n%5E2 is equivalent to 36m%5E2%2B30mn%2B30mn%2B25n%5E2)


note: %286m%2B5n%29%286m%2B5n%29 is equivalent to %286m%2B5n%29%5E2 since the term 6m%2B5n occurs twice. So 36m%5E2%2B60mn%2B25n%5E2 also factors to %286m%2B5n%29%5E2



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Answer:
So 36m%5E2%2B60mn%2B25n%5E2 factors to %286m%2B5n%29%5E2