SOLUTION: explain how to completely factor 3x2 - 3x - 60

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Question 626500: explain how to completely factor 3x2 - 3x - 60
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
3x%5E2-3x-60 Start with the given expression.


3%28x%5E2-x-20%29 Factor out the GCF 3.


Now let's try to factor the inner expression x%5E2-x-20


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


Now multiply the first coefficient 1 by the last term -20 to get %281%29%28-20%29=-20.


Now the question is: what two whole numbers multiply to -20 (the previous product) and add to the second coefficient -1?


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 -1:


First NumberSecond NumberSum
1-201+(-20)=-19
2-102+(-10)=-8
4-54+(-5)=-1
-120-1+20=19
-210-2+10=8
-45-4+5=1



From the table, we can see that the two numbers 4 and -5 add to -1 (the middle coefficient).


So the two numbers 4 and -5 both multiply to -20 and add to -1


Now replace the middle term -1x with 4x-5x. Remember, 4 and -5 add to -1. So this shows us that 4x-5x=-1x.


x%5E2%2Bhighlight%284x-5x%29-20 Replace the second term -1x with 4x-5x.


%28x%5E2%2B4x%29%2B%28-5x-20%29 Group the terms into two pairs.


x%28x%2B4%29%2B%28-5x-20%29 Factor out the GCF x from the first group.


x%28x%2B4%29-5%28x%2B4%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.


%28x-5%29%28x%2B4%29 Combine like terms. Or factor out the common term x%2B4


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So 3%28x%5E2-x-20%29 then factors further to 3%28x-5%29%28x%2B4%29


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Answer:


So 3x%5E2-3x-60 completely factors to 3%28x-5%29%28x%2B4%29.


In other words, 3x%5E2-3x-60=3%28x-5%29%28x%2B4%29.


Note: you can check the answer by expanding 3%28x-5%29%28x%2B4%29 to get 3x%5E2-3x-60 or by graphing the original expression and the answer (the two graphs should be identical).