SOLUTION: Factor completely. If a polynomial is prime, state this. 6x2 + 19x + 10

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Question 550523: Factor completely. If a polynomial is prime, state this.
6x2 + 19x + 10
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Looking at the expression 6x%5E2%2B19x%2B10, we can see that the first coefficient is 6, the second coefficient is 19, and the last term is 10.


Now multiply the first coefficient 6 by the last term 10 to get %286%29%2810%29=60.


Now the question is: what two whole numbers multiply to 60 (the previous product) and add to the second coefficient 19?


To find these two numbers, we need to list all of the factors of 60 (the previous product).


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


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


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 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 the table, we can see that the two numbers 4 and 15 add to 19 (the middle coefficient).


So the two numbers 4 and 15 both multiply to 60 and add to 19


Now replace the middle term 19x with 4x%2B15x. Remember, 4 and 15 add to 19. So this shows us that 4x%2B15x=19x.


6x%5E2%2Bhighlight%284x%2B15x%29%2B10 Replace the second term 19x with 4x%2B15x.


%286x%5E2%2B4x%29%2B%2815x%2B10%29 Group the terms into two pairs.


2x%283x%2B2%29%2B%2815x%2B10%29 Factor out the GCF 2x from the first group.


2x%283x%2B2%29%2B5%283x%2B2%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%2B5%29%283x%2B2%29 Combine like terms. Or factor out the common term 3x%2B2


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Answer:


So 6x%5E2%2B19x%2B10 factors to %282x%2B5%29%283x%2B2%29.


In other words, 6x%5E2%2B19x%2B10=%282x%2B5%29%283x%2B2%29.


Note: you can check the answer by expanding %282x%2B5%29%283x%2B2%29 to get 6x%5E2%2B19x%2B10 or by graphing the original expression and the answer (the two graphs should be identical).