SOLUTION: Directions are to factor each polynomial. If the polynomial cannot be factored using integers, write prime.
36m^2+60mn+25n^2
Algebra.Com
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) (Show Source): You can put this solution on YOUR website!
Looking at we can see that the first term is and the last term is 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 Number | Second Number | Sum | | 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 |
| -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
------------------------------------------------------------
Answer:
So factors to
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