SOLUTION: please help with this polynomial 10x^2+23x+12 thanks

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Question 185528: please help with this polynomial

10x^2+23x+12

thanks
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I assume that you want to factor.



Looking at the expression 10x%5E2%2B23x%2B12, we can see that the first coefficient is 10, the second coefficient is 23, and the last term is 12.


Now multiply the first coefficient 10 by the last term 12 to get %2810%29%2812%29=120.


Now the question is: what two whole numbers multiply to 120 (the previous product) and add to the second coefficient 23?


To find these two numbers, we need to list all of the factors of 120 (the previous product).


Factors of 120:
1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120
-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-20,-24,-30,-40,-60,-120


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


These factors pair up and multiply to 120.
1*120
2*60
3*40
4*30
5*24
6*20
8*15
10*12
(-1)*(-120)
(-2)*(-60)
(-3)*(-40)
(-4)*(-30)
(-5)*(-24)
(-6)*(-20)
(-8)*(-15)
(-10)*(-12)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 23:


First NumberSecond NumberSum
11201+120=121
2602+60=62
3403+40=43
4304+30=34
5245+24=29
6206+20=26
8158+15=23
101210+12=22
-1-120-1+(-120)=-121
-2-60-2+(-60)=-62
-3-40-3+(-40)=-43
-4-30-4+(-30)=-34
-5-24-5+(-24)=-29
-6-20-6+(-20)=-26
-8-15-8+(-15)=-23
-10-12-10+(-12)=-22



From the table, we can see that the two numbers 8 and 15 add to 23 (the middle coefficient).


So the two numbers 8 and 15 both multiply to 120 and add to 23


Now replace the middle term 23x with 8x%2B15x. Remember, 8 and 15 add to 23. So this shows us that 8x%2B15x=23x.


10x%5E2%2Bhighlight%288x%2B15x%29%2B12 Replace the second term 23x with 8x%2B15x.


%2810x%5E2%2B8x%29%2B%2815x%2B12%29 Group the terms into two pairs.


2x%285x%2B4%29%2B%2815x%2B12%29 Factor out the GCF 2x from the first group.


2x%285x%2B4%29%2B3%285x%2B4%29 Factor out 3 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%2B3%29%285x%2B4%29 Combine like terms. Or factor out the common term 5x%2B4

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Answer:


So 10x%5E2%2B23x%2B12 factors to %282x%2B3%29%285x%2B4%29.


Note: you can check the answer by FOILing %282x%2B3%29%285x%2B4%29 to get 10x%5E2%2B23x%2B12 or by graphing the original expression and the answer (the two graphs should be identical).