SOLUTION: 6y^3+25y^2+14y

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Question 539461: 6y^3+25y^2+14y
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

6y%5E3%2B25y%5E2%2B14y Start with the given expression.


y%286y%5E2%2B25y%2B14%29 Factor out the GCF y.


Now let's try to factor the inner expression 6y%5E2%2B25y%2B14


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Looking at the expression 6y%5E2%2B25y%2B14, we can see that the first coefficient is 6, the second coefficient is 25, and the last term is 14.


Now multiply the first coefficient 6 by the last term 14 to get %286%29%2814%29=84.


Now the question is: what two whole numbers multiply to 84 (the previous product) and add to the second coefficient 25?


To find these two numbers, we need to list all of the factors of 84 (the previous product).


Factors of 84:
1,2,3,4,6,7,12,14,21,28,42,84
-1,-2,-3,-4,-6,-7,-12,-14,-21,-28,-42,-84


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


These factors pair up and multiply to 84.
1*84 = 84
2*42 = 84
3*28 = 84
4*21 = 84
6*14 = 84
7*12 = 84
(-1)*(-84) = 84
(-2)*(-42) = 84
(-3)*(-28) = 84
(-4)*(-21) = 84
(-6)*(-14) = 84
(-7)*(-12) = 84

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 25:


First NumberSecond NumberSum
1841+84=85
2422+42=44
3283+28=31
4214+21=25
6146+14=20
7127+12=19
-1-84-1+(-84)=-85
-2-42-2+(-42)=-44
-3-28-3+(-28)=-31
-4-21-4+(-21)=-25
-6-14-6+(-14)=-20
-7-12-7+(-12)=-19



From the table, we can see that the two numbers 4 and 21 add to 25 (the middle coefficient).


So the two numbers 4 and 21 both multiply to 84 and add to 25


Now replace the middle term 25y with 4y%2B21y. Remember, 4 and 21 add to 25. So this shows us that 4y%2B21y=25y.


6y%5E2%2Bhighlight%284y%2B21y%29%2B14 Replace the second term 25y with 4y%2B21y.


%286y%5E2%2B4y%29%2B%2821y%2B14%29 Group the terms into two pairs.


2y%283y%2B2%29%2B%2821y%2B14%29 Factor out the GCF 2y from the first group.


2y%283y%2B2%29%2B7%283y%2B2%29 Factor out 7 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.


%282y%2B7%29%283y%2B2%29 Combine like terms. Or factor out the common term 3y%2B2


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So y%286y%5E2%2B25y%2B14%29 then factors further to y%282y%2B7%29%283y%2B2%29


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Answer:


So 6y%5E3%2B25y%5E2%2B14y completely factors to y%282y%2B7%29%283y%2B2%29.


In other words, 6y%5E3%2B25y%5E2%2B14y=y%282y%2B7%29%283y%2B2%29.


Note: you can check the answer by expanding y%282y%2B7%29%283y%2B2%29 to get 6y%5E3%2B25y%5E2%2B14y 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

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Jim