SOLUTION: Factor Completely c^3 + 20c^2 + 100c

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Question 339939: Factor Completely

c^3 + 20c^2 + 100c
Found 2 solutions by jim_thompson5910, Fombitz:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

c%5E3%2B20c%5E2%2B100c Start with the given expression


c%28c%5E2%2B20c%2B100%29 Factor out the GCF c


Now let's focus on the inner expression c%5E2%2B20c%2B100




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Looking at 1c%5E2%2B20c%2B100 we can see that the first term is 1c%5E2 and the last term is 100 where the coefficients are 1 and 100 respectively.

Now multiply the first coefficient 1 and the last coefficient 100 to get 100. Now what two numbers multiply to 100 and add to the middle coefficient 20? Let's list all of the factors of 100:



Factors of 100:
1,2,4,5,10,20,25,50

-1,-2,-4,-5,-10,-20,-25,-50 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 100
1*100
2*50
4*25
5*20
10*10
(-1)*(-100)
(-2)*(-50)
(-4)*(-25)
(-5)*(-20)
(-10)*(-10)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to 20? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 20

First NumberSecond NumberSum
11001+100=101
2502+50=52
4254+25=29
5205+20=25
101010+10=20
-1-100-1+(-100)=-101
-2-50-2+(-50)=-52
-4-25-4+(-25)=-29
-5-20-5+(-20)=-25
-10-10-10+(-10)=-20



From this list we can see that 10 and 10 add up to 20 and multiply to 100


Now looking at the expression 1c%5E2%2B20c%2B100, replace 20c with 10c%2B10c (notice 10c%2B10c adds up to 20c. So it is equivalent to 20c)

1c%5E2%2Bhighlight%2810c%2B10c%29%2B100


Now let's factor 1c%5E2%2B10c%2B10c%2B100 by grouping:


%281c%5E2%2B10c%29%2B%2810c%2B100%29 Group like terms


c%28c%2B10%29%2B10%28c%2B10%29 Factor out the GCF of c out of the first group. Factor out the GCF of 10 out of the second group


%28c%2B10%29%28c%2B10%29 Since we have a common term of c%2B10, we can combine like terms

So 1c%5E2%2B10c%2B10c%2B100 factors to %28c%2B10%29%28c%2B10%29


So this also means that 1c%5E2%2B20c%2B100 factors to %28c%2B10%29%28c%2B10%29 (since 1c%5E2%2B20c%2B100 is equivalent to 1c%5E2%2B10c%2B10c%2B100)


note: %28c%2B10%29%28c%2B10%29 is equivalent to %28c%2B10%29%5E2 since the term c%2B10 occurs twice. So 1c%5E2%2B20c%2B100 also factors to %28c%2B10%29%5E2



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So our expression goes from c%28c%5E2%2B20c%2B100%29 and factors further to c%28c%2B10%29%5E2


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Answer:

So c%5E3%2B20c%5E2%2B100c factors to c%28c%2B10%29%5E2

If you need more help, email me at jim_thompson5910@hotmail.com

Also, feel free to check out my tutoring website

Jim

Answer by Fombitz(32388) About Me  (Show Source):