SOLUTION: Can these problems be worked with the AC Method?? 3x^3y^2-3x^2y^2+3xy^2 35x^2-2x-1 Thank you

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Question 707276: Can these problems be worked with the AC Method??
3x^3y^2-3x^2y^2+3xy^2



35x^2-2x-1

Thank you
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 1



3x%5E3y%5E2-3x%5E2y%5E2%2B3xy%5E2 Start with the given expression.


3xy%5E2%28x%5E2-x%2B1%29 Factor out the GCF 3xy%5E2.


Now let's try to factor the inner expression x%5E2-x%2B1


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Looking at the expression x%5E2-x%2B1, we can see that the first coefficient is 1, the second coefficient is -1, and the last term is 1.


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


Now the question is: what two whole numbers multiply to 1 (the previous product) and add to the second coefficient -1?


To find these two numbers, we need to list all of the factors of 1 (the previous product).


Factors of 1:
1
-1


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


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -1:


First NumberSecond NumberSum
111+1=2
-1-1-1+(-1)=-2



From the table, we can see that there are no pairs of numbers which add to -1. So x%5E2-x%2B1 cannot be factored.


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Answer:


So 3x%5E3y%5E2-3x%5E2y%5E2%2B3xy%5E2 simply factors to 3xy%5E2%28x%5E2-x%2B1%29


In other words, 3x%5E3y%5E2-3x%5E2y%5E2%2B3xy%5E2=3xy%5E2%28x%5E2-x%2B1%29.

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# 2



Looking at the expression 35x%5E2-2x-1, we can see that the first coefficient is 35, the second coefficient is -2, and the last term is -1.


Now multiply the first coefficient 35 by the last term -1 to get %2835%29%28-1%29=-35.


Now the question is: what two whole numbers multiply to -35 (the previous product) and add to the second coefficient -2?


To find these two numbers, we need to list all of the factors of -35 (the previous product).


Factors of -35:
1,5,7,35
-1,-5,-7,-35


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


These factors pair up and multiply to -35.
1*(-35) = -35
5*(-7) = -35
(-1)*(35) = -35
(-5)*(7) = -35

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


First NumberSecond NumberSum
1-351+(-35)=-34
5-75+(-7)=-2
-135-1+35=34
-57-5+7=2



From the table, we can see that the two numbers 5 and -7 add to -2 (the middle coefficient).


So the two numbers 5 and -7 both multiply to -35 and add to -2


Now replace the middle term -2x with 5x-7x. Remember, 5 and -7 add to -2. So this shows us that 5x-7x=-2x.


35x%5E2%2Bhighlight%285x-7x%29-1 Replace the second term -2x with 5x-7x.


%2835x%5E2%2B5x%29%2B%28-7x-1%29 Group the terms into two pairs.


5x%287x%2B1%29%2B%28-7x-1%29 Factor out the GCF 5x from the first group.


5x%287x%2B1%29-1%287x%2B1%29 Factor out 1 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.


%285x-1%29%287x%2B1%29 Combine like terms. Or factor out the common term 7x%2B1


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Answer:


So 35x%5E2-2x-1 factors to %285x-1%29%287x%2B1%29.


In other words, 35x%5E2-2x-1=%285x-1%29%287x%2B1%29.


Note: you can check the answer by expanding %285x-1%29%287x%2B1%29 to get 35x%5E2-2x-1 or by graphing the original expression and the answer (the two graphs should be identical).