SOLUTION: how do you factor {{{x^2 -22x+120}}}

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Question 913154: how do you factor x%5E2+-22x%2B120
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression x%5E2-22x%2B120, we can see that the first coefficient is 1, the second coefficient is -22, and the last term is 120.


Now multiply the first coefficient 1 by the last term 120 to get %281%29%28120%29=120.


Now the question is: what two whole numbers multiply to 120 (the previous product) and add to the second coefficient -22?


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 -22:


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 -10 and -12 add to -22 (the middle coefficient).


So the two numbers -10 and -12 both multiply to 120 and add to -22


Now replace the middle term -22x with -10x-12x. Remember, -10 and -12 add to -22. So this shows us that -10x-12x=-22x.


x%5E2%2Bhighlight%28-10x-12x%29%2B120 Replace the second term -22x with -10x-12x.


%28x%5E2-10x%29%2B%28-12x%2B120%29 Group the terms into two pairs.


x%28x-10%29%2B%28-12x%2B120%29 Factor out the GCF x from the first group.


x%28x-10%29-12%28x-10%29 Factor out 12 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-12%29%28x-10%29 Combine like terms. Or factor out the common term x-10


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Answer:


So x%5E2-22x%2B120 factors to %28x-12%29%28x-10%29.


In other words, x%5E2-22x%2B120=%28x-12%29%28x-10%29.


Note: you can check the answer by expanding %28x-12%29%28x-10%29 to get x%5E2-22x%2B120 or by graphing the original expression and the answer (the two graphs should be identical).


Let me know if you need more help or if you need me to explain a step in more detail.
Feel free to email me at jim_thompson5910@hotmail.com
or you can visit my website here: http://www.freewebs.com/jimthompson5910/home.html

Thanks,

Jim