SOLUTION: Factor x^2 - 19x - 120

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Question 626497: Factor x^2 - 19x - 120
Answer by jim_thompson5910(35256) About Me  (Show Source):
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Looking at the expression x%5E2-19x-120, we can see that the first coefficient is 1, the second coefficient is -19, and the last term is -120.


Now multiply the first coefficient 1 by the last term -120 to get %281%29%28-120%29=-120.


Now the question is: what two whole numbers multiply to -120 (the previous product) and add to the second coefficient -19?


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) = -120
2*(-60) = -120
3*(-40) = -120
4*(-30) = -120
5*(-24) = -120
6*(-20) = -120
8*(-15) = -120
10*(-12) = -120
(-1)*(120) = -120
(-2)*(60) = -120
(-3)*(40) = -120
(-4)*(30) = -120
(-5)*(24) = -120
(-6)*(20) = -120
(-8)*(15) = -120
(-10)*(12) = -120

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -19:


First NumberSecond NumberSum
1-1201+(-120)=-119
2-602+(-60)=-58
3-403+(-40)=-37
4-304+(-30)=-26
5-245+(-24)=-19
6-206+(-20)=-14
8-158+(-15)=-7
10-1210+(-12)=-2
-1120-1+120=119
-260-2+60=58
-340-3+40=37
-430-4+30=26
-524-5+24=19
-620-6+20=14
-815-8+15=7
-1012-10+12=2



From the table, we can see that the two numbers 5 and -24 add to -19 (the middle coefficient).


So the two numbers 5 and -24 both multiply to -120 and add to -19


Now replace the middle term -19x with 5x-24x. Remember, 5 and -24 add to -19. So this shows us that 5x-24x=-19x.


x%5E2%2Bhighlight%285x-24x%29-120 Replace the second term -19x with 5x-24x.


%28x%5E2%2B5x%29%2B%28-24x-120%29 Group the terms into two pairs.


x%28x%2B5%29%2B%28-24x-120%29 Factor out the GCF x from the first group.


x%28x%2B5%29-24%28x%2B5%29 Factor out 24 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-24%29%28x%2B5%29 Combine like terms. Or factor out the common term x%2B5


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Answer:


So x%5E2-19x-120 factors to %28x-24%29%28x%2B5%29.


In other words, x%5E2-19x-120=%28x-24%29%28x%2B5%29.


Note: you can check the answer by expanding %28x-24%29%28x%2B5%29 to get x%5E2-19x-120 or by graphing the original expression and the answer (the two graphs should be identical).

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