SOLUTION: Can someone please help me? Factor and show work, If prime state this. 144-25t+t^2

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Question 405579: Can someone please help me?
Factor and show work, If prime state this.
144-25t+t^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Rearrange the terms to get t%5E2-25t%2B144




Looking at t%5E2-25t%2B144 we can see that the first term is t%5E2 and the last term is 144 where the coefficients are 1 and 144 respectively.

Now multiply the first coefficient 1 and the last coefficient 144 to get 144. Now what two numbers multiply to 144 and add to the middle coefficient -25? Let's list all of the factors of 144:



Factors of 144:
1,2,3,4,6,8,9,12,16,18,24,36,48,72

-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 144
1*144
2*72
3*48
4*36
6*24
8*18
9*16
12*12
(-1)*(-144)
(-2)*(-72)
(-3)*(-48)
(-4)*(-36)
(-6)*(-24)
(-8)*(-18)
(-9)*(-16)
(-12)*(-12)

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


Now which of these pairs add to -25? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -25

First NumberSecond NumberSum
11441+144=145
2722+72=74
3483+48=51
4364+36=40
6246+24=30
8188+18=26
9169+16=25
121212+12=24
-1-144-1+(-144)=-145
-2-72-2+(-72)=-74
-3-48-3+(-48)=-51
-4-36-4+(-36)=-40
-6-24-6+(-24)=-30
-8-18-8+(-18)=-26
-9-16-9+(-16)=-25
-12-12-12+(-12)=-24



From this list we can see that -9 and -16 add up to -25 and multiply to 144


Now looking at the expression t%5E2-25t%2B144, replace -25t with -9t-16t (notice -9t-16t adds up to -25t. So it is equivalent to -25t)

t%5E2%2Bhighlight%28-9t-16t%29%2B144


Now let's factor t%5E2-9t-16t%2B144 by grouping:


%28t%5E2-9t%29%2B%28-16t%2B144%29 Group like terms


t%28t-9%29-16%28t-9%29 Factor out the GCF of t out of the first group. Factor out the GCF of -16 out of the second group


%28t-16%29%28t-9%29 Since we have a common term of t-9, we can combine like terms

So t%5E2-9t-16t%2B144 factors to %28t-16%29%28t-9%29


So this also means that t%5E2-25t%2B144 factors to %28t-16%29%28t-9%29 (since t%5E2-25t%2B144 is equivalent to t%5E2-9t-16t%2B144)



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Answer:
So t%5E2-25t%2B144 factors to %28t-16%29%28t-9%29


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Jim