SOLUTION: 50x^3+95x^2+30x Factor the trinomial completely

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Question 191478: 50x^3+95x^2+30x
Factor the trinomial completely
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

50x%5E3%2B95x%5E2%2B30x Start with the given expression


5x%2810x%5E2%2B19x%2B6%29 Factor out the GCF 5x


Now let's focus on the inner expression 10x%5E2%2B19x%2B6


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Looking at 10x%5E2%2B19x%2B6 we can see that the first term is 10x%5E2 and the last term is 6 where the coefficients are 10 and 6 respectively.

Now multiply the first coefficient 10 and the last coefficient 6 to get 60. Now what two numbers multiply to 60 and add to the middle coefficient 19? Let's list all of the factors of 60:



Factors of 60:
1,2,3,4,5,6,10,12,15,20,30,60

-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 60
1*60
2*30
3*20
4*15
5*12
6*10
(-1)*(-60)
(-2)*(-30)
(-3)*(-20)
(-4)*(-15)
(-5)*(-12)
(-6)*(-10)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to 19? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 19

First NumberSecond NumberSum
1601+60=61
2302+30=32
3203+20=23
4154+15=19
5125+12=17
6106+10=16
-1-60-1+(-60)=-61
-2-30-2+(-30)=-32
-3-20-3+(-20)=-23
-4-15-4+(-15)=-19
-5-12-5+(-12)=-17
-6-10-6+(-10)=-16



From this list we can see that 4 and 15 add up to 19 and multiply to 60


Now looking at the expression 10x%5E2%2B19x%2B6, replace 19x with 4x%2B15x (notice 4x%2B15x adds up to 19x. So it is equivalent to 19x)


10x%5E2%2Bhighlight%284x%2B15x%29%2B6


Now let's factor 10x%5E2%2B4x%2B15x%2B6 by grouping:


%2810x%5E2%2B4x%29%2B%2815x%2B6%29 Group like terms


2x%285x%2B2%29%2B3%285x%2B2%29 Factor out the GCF of 2x out of the first group. Factor out the GCF of 3 out of the second group


%282x%2B3%29%285x%2B2%29 Since we have a common term of 5x%2B2, we can combine like terms


So 10x%5E2%2B4x%2B15x%2B6 factors to %282x%2B3%29%285x%2B2%29


So this also means that 10x%5E2%2B19x%2B6 factors to %282x%2B3%29%285x%2B2%29 (since 10x%5E2%2B19x%2B6 is equivalent to 10x%5E2%2B4x%2B15x%2B6)



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So our expression goes from 5x%2810x%5E2%2B19x%2B6%29 and factors further to 5x%282x%2B3%29%285x%2B2%29


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Answer:

So 50x%5E3%2B95x%5E2%2B30x completely factors to 5x%282x%2B3%29%285x%2B2%29