SOLUTION: 3x^2-36x+60. Can you help me solve this as a grouping polynomial.

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Question 597922: 3x^2-36x+60. Can you help me solve this as a grouping polynomial.

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

3x%5E2-36x%2B60 Start with the given expression.


3%28x%5E2-12x%2B20%29 Factor out the GCF 3.


Now let's try to factor the inner expression x%5E2-12x%2B20


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Looking at the expression x%5E2-12x%2B20, we can see that the first coefficient is 1, the second coefficient is -12, and the last term is 20.


Now multiply the first coefficient 1 by the last term 20 to get %281%29%2820%29=20.


Now the question is: what two whole numbers multiply to 20 (the previous product) and add to the second coefficient -12?


To find these two numbers, we need to list all of the factors of 20 (the previous product).


Factors of 20:
1,2,4,5,10,20
-1,-2,-4,-5,-10,-20


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


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -12:


First NumberSecond NumberSum
1201+20=21
2102+10=12
454+5=9
-1-20-1+(-20)=-21
-2-10-2+(-10)=-12
-4-5-4+(-5)=-9



From the table, we can see that the two numbers -2 and -10 add to -12 (the middle coefficient).


So the two numbers -2 and -10 both multiply to 20 and add to -12


Now replace the middle term -12x with -2x-10x. Remember, -2 and -10 add to -12. So this shows us that -2x-10x=-12x.


x%5E2%2Bhighlight%28-2x-10x%29%2B20 Replace the second term -12x with -2x-10x.


%28x%5E2-2x%29%2B%28-10x%2B20%29 Group the terms into two pairs.


x%28x-2%29%2B%28-10x%2B20%29 Factor out the GCF x from the first group.


x%28x-2%29-10%28x-2%29 Factor out 10 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.


%28x-10%29%28x-2%29 Combine like terms. Or factor out the common term x-2


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So 3%28x%5E2-12x%2B20%29 then factors further to 3%28x-10%29%28x-2%29


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Answer:


So 3x%5E2-36x%2B60 completely factors to 3%28x-10%29%28x-2%29.


In other words, 3x%5E2-36x%2B60=3%28x-10%29%28x-2%29.


Note: you can check the answer by expanding 3%28x-10%29%28x-2%29 to get 3x%5E2-36x%2B60 or by graphing the original expression and the answer (the two graphs should be identical).