SOLUTION: I need to completely factor 6x^3-31x^2+5x.

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Question 641543: I need to completely factor 6x^3-31x^2+5x.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

6x%5E3-31x%5E2%2B5x Start with the given expression.


x%286x%5E2-31x%2B5%29 Factor out the GCF x.


Now let's try to factor the inner expression 6x%5E2-31x%2B5


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Looking at the expression 6x%5E2-31x%2B5, we can see that the first coefficient is 6, the second coefficient is -31, and the last term is 5.


Now multiply the first coefficient 6 by the last term 5 to get %286%29%285%29=30.


Now the question is: what two whole numbers multiply to 30 (the previous product) and add to the second coefficient -31?


To find these two numbers, we need to list all of the factors of 30 (the previous product).


Factors of 30:
1,2,3,5,6,10,15,30
-1,-2,-3,-5,-6,-10,-15,-30


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


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

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


First NumberSecond NumberSum
1301+30=31
2152+15=17
3103+10=13
565+6=11
-1-30-1+(-30)=-31
-2-15-2+(-15)=-17
-3-10-3+(-10)=-13
-5-6-5+(-6)=-11



From the table, we can see that the two numbers -1 and -30 add to -31 (the middle coefficient).


So the two numbers -1 and -30 both multiply to 30 and add to -31


Now replace the middle term -31x with -x-30x. Remember, -1 and -30 add to -31. So this shows us that -x-30x=-31x.


6x%5E2%2Bhighlight%28-x-30x%29%2B5 Replace the second term -31x with -x-30x.


%286x%5E2-x%29%2B%28-30x%2B5%29 Group the terms into two pairs.


x%286x-1%29%2B%28-30x%2B5%29 Factor out the GCF x from the first group.


x%286x-1%29-5%286x-1%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%286x-1%29 Combine like terms. Or factor out the common term 6x-1


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So x%286x%5E2-31x%2B5%29 then factors further to x%28x-5%29%286x-1%29


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Answer:


So 6x%5E3-31x%5E2%2B5x completely factors to x%28x-5%29%286x-1%29.


In other words, 6x%5E3-31x%5E2%2B5x=x%28x-5%29%286x-1%29.


Note: you can check the answer by expanding x%28x-5%29%286x-1%29 to get 6x%5E3-31x%5E2%2B5x or by graphing the original expression and the answer (the two graphs should be identical).

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