SOLUTION: Factor 3xy^2 - 6xy - 45x

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Question 527931: Factor 3xy^2 - 6xy - 45x
Answer by jim_thompson5910(35256) About Me  (Show Source):
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3xy%5E2-6xy-45x Start with the given expression.


3x%28y%5E2-2y-15%29 Factor out the GCF 3x.


Now let's try to factor the inner expression y%5E2-2y-15


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Looking at the expression y%5E2-2y-15, we can see that the first coefficient is 1, the second coefficient is -2, and the last term is -15.


Now multiply the first coefficient 1 by the last term -15 to get %281%29%28-15%29=-15.


Now the question is: what two whole numbers multiply to -15 (the previous product) and add to the second coefficient -2?


To find these two numbers, we need to list all of the factors of -15 (the previous product).


Factors of -15:
1,3,5,15
-1,-3,-5,-15


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


These factors pair up and multiply to -15.
1*(-15) = -15
3*(-5) = -15
(-1)*(15) = -15
(-3)*(5) = -15

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -2:


First NumberSecond NumberSum
1-151+(-15)=-14
3-53+(-5)=-2
-115-1+15=14
-35-3+5=2



From the table, we can see that the two numbers 3 and -5 add to -2 (the middle coefficient).


So the two numbers 3 and -5 both multiply to -15 and add to -2


Now replace the middle term -2y with 3y-5y. Remember, 3 and -5 add to -2. So this shows us that 3y-5y=-2y.


y%5E2%2Bhighlight%283y-5y%29-15 Replace the second term -2y with 3y-5y.


%28y%5E2%2B3y%29%2B%28-5y-15%29 Group the terms into two pairs.


y%28y%2B3%29%2B%28-5y-15%29 Factor out the GCF y from the first group.


y%28y%2B3%29-5%28y%2B3%29 Factor out 5 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.


%28y-5%29%28y%2B3%29 Combine like terms. Or factor out the common term y%2B3


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So 3x%28y%5E2-2y-15%29 then factors further to 3x%28y-5%29%28y%2B3%29


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Answer:


So 3xy%5E2-6xy-45x completely factors to 3x%28y-5%29%28y%2B3%29.


In other words, 3xy%5E2-6xy-45x=3x%28y-5%29%28y%2B3%29.


Note: you can check the answer by expanding 3x%28y-5%29%28y%2B3%29 to get 3xy%5E2-6xy-45x 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