SOLUTION: factoring trinomials completely..........this is my problem and I can't figure out how to work any of them..... 36t^2+390t+64 (?t+?)(?t+?)

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: factoring trinomials completely..........this is my problem and I can't figure out how to work any of them..... 36t^2+390t+64 (?t+?)(?t+?)      Log On


   

Question 207386: factoring trinomials completely..........this is my problem and I can't figure out how to work any of them.....
36t^2+390t+64
(?t+?)(?t+?)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!


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36t%5E2%2B390t%2B64 Start with the given expression.


2%2818t%5E2%2B195t%2B32%29 Factor out the GCF 2.


Now let's try to factor the inner expression 18t%5E2%2B195t%2B32


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Looking at the expression 18t%5E2%2B195t%2B32, we can see that the first coefficient is 18, the second coefficient is 195, and the last term is 32.


Now multiply the first coefficient 18 by the last term 32 to get %2818%29%2832%29=576.


Now the question is: what two whole numbers multiply to 576 (the previous product) and add to the second coefficient 195?


To find these two numbers, we need to list all of the factors of 576 (the previous product).


Factors of 576:
1,2,3,4,6,8,9,12,16,18,24,32,36,48,64,72,96,144,192,288,576
-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-32,-36,-48,-64,-72,-96,-144,-192,-288,-576


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to 576.
1*576 = 576
2*288 = 576
3*192 = 576
4*144 = 576
6*96 = 576
8*72 = 576
9*64 = 576
12*48 = 576
16*36 = 576
18*32 = 576
24*24 = 576
(-1)*(-576) = 576
(-2)*(-288) = 576
(-3)*(-192) = 576
(-4)*(-144) = 576
(-6)*(-96) = 576
(-8)*(-72) = 576
(-9)*(-64) = 576
(-12)*(-48) = 576
(-16)*(-36) = 576
(-18)*(-32) = 576
(-24)*(-24) = 576

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 195:


First NumberSecond NumberSum
15761+576=577
22882+288=290
31923+192=195
41444+144=148
6966+96=102
8728+72=80
9649+64=73
124812+48=60
163616+36=52
183218+32=50
242424+24=48
-1-576-1+(-576)=-577
-2-288-2+(-288)=-290
-3-192-3+(-192)=-195
-4-144-4+(-144)=-148
-6-96-6+(-96)=-102
-8-72-8+(-72)=-80
-9-64-9+(-64)=-73
-12-48-12+(-48)=-60
-16-36-16+(-36)=-52
-18-32-18+(-32)=-50
-24-24-24+(-24)=-48



From the table, we can see that the two numbers 3 and 192 add to 195 (the middle coefficient).


So the two numbers 3 and 192 both multiply to 576 and add to 195


Now replace the middle term 195t with 3t%2B192t. Remember, 3 and 192 add to 195. So this shows us that 3t%2B192t=195t.


18t%5E2%2Bhighlight%283t%2B192t%29%2B32 Replace the second term 195t with 3t%2B192t.


%2818t%5E2%2B3t%29%2B%28192t%2B32%29 Group the terms into two pairs.


3t%286t%2B1%29%2B%28192t%2B32%29 Factor out the GCF 3t from the first group.


3t%286t%2B1%29%2B32%286t%2B1%29 Factor out 32 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


%283t%2B32%29%286t%2B1%29 Combine like terms. Or factor out the common term 6t%2B1


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So 2%2818t%5E2%2B195t%2B32%29 then factors further to 2%283t%2B32%29%286t%2B1%29


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Answer:


So 36t%5E2%2B390t%2B64 completely factors to 2%283t%2B32%29%286t%2B1%29.


In other words, 36t%5E2%2B390t%2B64=2%283t%2B32%29%286t%2B1%29.


Note: you can check the answer by expanding 2%283t%2B32%29%286t%2B1%29 to get 36t%5E2%2B390t%2B64 or by graphing the original expression and the answer (the two graphs should be identical).

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