SOLUTION: factor the trinomial by grouping : 15x^3+47x^2+28x

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Question 527181: factor the trinomial by grouping : 15x^3+47x^2+28x
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

15x%5E3%2B47x%5E2%2B28x Start with the given expression.


x%2815x%5E2%2B47x%2B28%29 Factor out the GCF x.


Now let's try to factor the inner expression 15x%5E2%2B47x%2B28


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Looking at the expression 15x%5E2%2B47x%2B28, we can see that the first coefficient is 15, the second coefficient is 47, and the last term is 28.


Now multiply the first coefficient 15 by the last term 28 to get %2815%29%2828%29=420.


Now the question is: what two whole numbers multiply to 420 (the previous product) and add to the second coefficient 47?


To find these two numbers, we need to list all of the factors of 420 (the previous product).


Factors of 420:
1,2,3,4,5,6,7,10,12,14,15,20,21,28,30,35,42,60,70,84,105,140,210,420
-1,-2,-3,-4,-5,-6,-7,-10,-12,-14,-15,-20,-21,-28,-30,-35,-42,-60,-70,-84,-105,-140,-210,-420


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


These factors pair up and multiply to 420.
1*420 = 420
2*210 = 420
3*140 = 420
4*105 = 420
5*84 = 420
6*70 = 420
7*60 = 420
10*42 = 420
12*35 = 420
14*30 = 420
15*28 = 420
20*21 = 420
(-1)*(-420) = 420
(-2)*(-210) = 420
(-3)*(-140) = 420
(-4)*(-105) = 420
(-5)*(-84) = 420
(-6)*(-70) = 420
(-7)*(-60) = 420
(-10)*(-42) = 420
(-12)*(-35) = 420
(-14)*(-30) = 420
(-15)*(-28) = 420
(-20)*(-21) = 420

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 47:


First NumberSecond NumberSum
14201+420=421
22102+210=212
31403+140=143
41054+105=109
5845+84=89
6706+70=76
7607+60=67
104210+42=52
123512+35=47
143014+30=44
152815+28=43
202120+21=41
-1-420-1+(-420)=-421
-2-210-2+(-210)=-212
-3-140-3+(-140)=-143
-4-105-4+(-105)=-109
-5-84-5+(-84)=-89
-6-70-6+(-70)=-76
-7-60-7+(-60)=-67
-10-42-10+(-42)=-52
-12-35-12+(-35)=-47
-14-30-14+(-30)=-44
-15-28-15+(-28)=-43
-20-21-20+(-21)=-41



From the table, we can see that the two numbers 12 and 35 add to 47 (the middle coefficient).


So the two numbers 12 and 35 both multiply to 420 and add to 47


Now replace the middle term 47x with 12x%2B35x. Remember, 12 and 35 add to 47. So this shows us that 12x%2B35x=47x.


15x%5E2%2Bhighlight%2812x%2B35x%29%2B28 Replace the second term 47x with 12x%2B35x.


%2815x%5E2%2B12x%29%2B%2835x%2B28%29 Group the terms into two pairs.


3x%285x%2B4%29%2B%2835x%2B28%29 Factor out the GCF 3x from the first group.


3x%285x%2B4%29%2B7%285x%2B4%29 Factor out 7 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.


%283x%2B7%29%285x%2B4%29 Combine like terms. Or factor out the common term 5x%2B4


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So x%2815x%5E2%2B47x%2B28%29 then factors further to x%283x%2B7%29%285x%2B4%29


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Answer:


So 15x%5E3%2B47x%5E2%2B28x completely factors to x%283x%2B7%29%285x%2B4%29.


In other words, 15x%5E3%2B47x%5E2%2B28x=x%283x%2B7%29%285x%2B4%29.


Note: you can check the answer by expanding x%283x%2B7%29%285x%2B4%29 to get 15x%5E3%2B47x%5E2%2B28x or by graphing the original expression and the answer (the two graphs should be identical).



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Jim