SOLUTION: Factor completly (20b^2+100bt+125t^2) I understand the GCF is 5 and I had tried to factor it out getting (5(4b+5t)(b^2+25t)) No matter where I place the numbers or what

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Factor completly (20b^2+100bt+125t^2) I understand the GCF is 5 and I had tried to factor it out getting (5(4b+5t)(b^2+25t)) No matter where I place the numbers or what       Log On


   

Question 387339: Factor completly
(20b^2+100bt+125t^2)
I understand the GCF is 5 and I had tried to factor it out getting
(5(4b+5t)(b^2+25t))
No matter where I place the numbers or what numbers I use I can't get both 100bt AND 125t^2. I e-mailed my teacher for help 5 days ago with still no response and at this point any help is REALLY appreciated!!!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
20b%5E2%2B100bt%2B125t%5E2 Start with the given expression.


5%284b%5E2%2B20bt%2B25t%5E2%29 Factor out the GCF 5.


Now let's try to factor the inner expression 4b%5E2%2B20bt%2B25t%5E2


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Looking at the expression 4b%5E2%2B20bt%2B25t%5E2, we can see that the first coefficient is 4, the second coefficient is 20, and the last coefficient is 25.


Now multiply the first coefficient 4 by the last coefficient 25 to get %284%29%2825%29=100.


Now the question is: what two whole numbers multiply to 100 (the previous product) and add to the second coefficient 20?


To find these two numbers, we need to list all of the factors of 100 (the previous product).


Factors of 100:
1,2,4,5,10,20,25,50,100
-1,-2,-4,-5,-10,-20,-25,-50,-100


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


These factors pair up and multiply to 100.
1*100 = 100
2*50 = 100
4*25 = 100
5*20 = 100
10*10 = 100
(-1)*(-100) = 100
(-2)*(-50) = 100
(-4)*(-25) = 100
(-5)*(-20) = 100
(-10)*(-10) = 100

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 20:


First NumberSecond NumberSum
11001+100=101
2502+50=52
4254+25=29
5205+20=25
101010+10=20
-1-100-1+(-100)=-101
-2-50-2+(-50)=-52
-4-25-4+(-25)=-29
-5-20-5+(-20)=-25
-10-10-10+(-10)=-20



From the table, we can see that the two numbers 10 and 10 add to 20 (the middle coefficient).


So the two numbers 10 and 10 both multiply to 100 and add to 20


Now replace the middle term 20bt with 10bt%2B10bt. Remember, 10 and 10 add to 20. So this shows us that 10bt%2B10bt=20bt.


4b%5E2%2Bhighlight%2810bt%2B10bt%29%2B25t%5E2 Replace the second term 20bt with 10bt%2B10bt.


%284b%5E2%2B10bt%29%2B%2810bt%2B25t%5E2%29 Group the terms into two pairs.


2b%282b%2B5t%29%2B%2810bt%2B25t%5E2%29 Factor out the GCF 2b from the first group.


2b%282b%2B5t%29%2B5t%282b%2B5t%29 Factor out 5t 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.


%282b%2B5t%29%282b%2B5t%29 Combine like terms. Or factor out the common term 2b%2B5t


%282b%2B5t%29%5E2 Condense the terms.


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So 5%284b%5E2%2B20bt%2B25t%5E2%29 then factors further to 5%282b%2B5t%29%5E2


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Answer:


So 20b%5E2%2B100bt%2B125t%5E2 completely factors to 5%282b%2B5t%29%5E2.


In other words, 20b%5E2%2B100bt%2B125t%5E2=5%282b%2B5t%29%5E2.


Note: you can check the answer by expanding 5%282b%2B5t%29%5E2 to get 20b%5E2%2B100bt%2B125t%5E2 or by graphing the original expression and the answer (the two graphs should be identical).


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

Also, feel free to check out my tutoring website

Jim