SOLUTION: Dear sir or mam:factoring trinomials completely 120p+100p^2+36

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Question 207781: Dear sir or mam:factoring trinomials completely
120p+100p^2+36
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

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120p%2B100p%5E2%2B36 Start with the given expression.


100p%5E2%2B120p%2B36 Rearrange the terms.


4%2825p%5E2%2B30p%2B9%29 Factor out the GCF 4.


Now let's try to factor the inner expression 25p%5E2%2B30p%2B9


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Looking at the expression 25p%5E2%2B30p%2B9, we can see that the first coefficient is 25, the second coefficient is 30, and the last term is 9.


Now multiply the first coefficient 25 by the last term 9 to get %2825%29%289%29=225.


Now the question is: what two whole numbers multiply to 225 (the previous product) and add to the second coefficient 30?


To find these two numbers, we need to list all of the factors of 225 (the previous product).


Factors of 225:
1,3,5,9,15,25,45,75,225
-1,-3,-5,-9,-15,-25,-45,-75,-225


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


These factors pair up and multiply to 225.
1*225 = 225
3*75 = 225
5*45 = 225
9*25 = 225
15*15 = 225
(-1)*(-225) = 225
(-3)*(-75) = 225
(-5)*(-45) = 225
(-9)*(-25) = 225
(-15)*(-15) = 225

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 30:


First NumberSecond NumberSum
12251+225=226
3753+75=78
5455+45=50
9259+25=34
151515+15=30
-1-225-1+(-225)=-226
-3-75-3+(-75)=-78
-5-45-5+(-45)=-50
-9-25-9+(-25)=-34
-15-15-15+(-15)=-30



From the table, we can see that the two numbers 15 and 15 add to 30 (the middle coefficient).


So the two numbers 15 and 15 both multiply to 225 and add to 30


Now replace the middle term 30p with 15p%2B15p. Remember, 15 and 15 add to 30. So this shows us that 15p%2B15p=30p.


25p%5E2%2Bhighlight%2815p%2B15p%29%2B9 Replace the second term 30p with 15p%2B15p.


%2825p%5E2%2B15p%29%2B%2815p%2B9%29 Group the terms into two pairs.


5p%285p%2B3%29%2B%2815p%2B9%29 Factor out the GCF 5p from the first group.


5p%285p%2B3%29%2B3%285p%2B3%29 Factor out 3 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.


%285p%2B3%29%285p%2B3%29 Combine like terms. Or factor out the common term 5p%2B3


%285p%2B3%29%5E2 Condense the terms.


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So 4%2825p%5E2%2B30p%2B9%29 then factors further to 4%285p%2B3%29%5E2


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Answer:


So 120p%2B100p%5E2%2B36 completely factors to 4%285p%2B3%29%5E2.


In other words, 120p%2B100p%5E2%2B36=4%285p%2B3%29%5E2.


Note: you can check the answer by expanding 4%285p%2B3%29%5E2 to get 120p%2B100p%5E2%2B36 or by graphing the original expression and the answer (the two graphs should be identical).

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