SOLUTION: factor completly: 10x^3-31x+15

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Question 153200: factor completly:
10x^3-31x+15
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Are you sure that the expression is not 10x%5E2-31x%2B15???


Looking at the expression 10x%5E2-31x%2B15, we can see that the first coefficient is 10, the second coefficient is -31, and the last term is 15.


Now multiply the first coefficient 10 by the last term 15 to get %2810%29%2815%29=150.


Now the question is: what two whole numbers multiply to 150 (the previous product) and add to the second coefficient -31?


To find these two numbers, we need to list all of the factors of 150 (the previous product).


Factors of 150:
1,2,3,5,6,10,15,25,30,50,75,150
-1,-2,-3,-5,-6,-10,-15,-25,-30,-50,-75,-150


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


These factors pair up and multiply to 150.
1*150
2*75
3*50
5*30
6*25
10*15
(-1)*(-150)
(-2)*(-75)
(-3)*(-50)
(-5)*(-30)
(-6)*(-25)
(-10)*(-15)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -31:


First NumberSecond NumberSum
11501+150=151
2752+75=77
3503+50=53
5305+30=35
6256+25=31
101510+15=25
-1-150-1+(-150)=-151
-2-75-2+(-75)=-77
-3-50-3+(-50)=-53
-5-30-5+(-30)=-35
-6-25-6+(-25)=-31
-10-15-10+(-15)=-25



From the table, we can see that the two numbers -6 and -25 add to -31 (the middle coefficient).


So the two numbers -6 and -25 both multiply to 150 and add to -31


Now replace the middle term -31x with -6x-25x. Remember, -6 and -25 add to -31. So this shows us that -6x-25x=-31x.


10x%5E2%2Bhighlight%28-6x-25x%29%2B15 Replace the second term -31x with -6x-25x.


%2810x%5E2-6x%29%2B%28-25x%2B15%29 Group the terms into two pairs.


2x%285x-3%29%2B%28-25x%2B15%29 Factor out the GCF 2x from the first group.


2x%285x-3%29-5%285x-3%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.


%282x-5%29%285x-3%29 Combine like terms. Or factor out the common term 5x-3

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Answer:


So 10x%5E2-31x%2B15 factors to %282x-5%29%285x-3%29.


Note: you can check the answer by FOILing %282x-5%29%285x-3%29 to get 10x%5E2-31x%2B15 or by graphing the original expression and the answer (the two graphs should be identical).