SOLUTION: Factor completely. If it is prime, say so. x^2+40x+400

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Question 109730: Factor completely. If it is prime, say so.
x^2+40x+400
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression x%5E2%2B40x%2B400, we can see that the first coefficient is 1, the second coefficient is 40, and the last term is 400.



Now multiply the first coefficient 1 by the last term 400 to get %281%29%28400%29=400.



Now the question is: what two whole numbers multiply to 400 (the previous product) and add to the second coefficient 40?



To find these two numbers, we need to list all of the factors of 400 (the previous product).



Factors of 400:

1,2,4,5,8,10,16,20,25,40,50,80,100,200,400

-1,-2,-4,-5,-8,-10,-16,-20,-25,-40,-50,-80,-100,-200,-400



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



These factors pair up and multiply to 400.

1*400 = 400
2*200 = 400
4*100 = 400
5*80 = 400
8*50 = 400
10*40 = 400
16*25 = 400
20*20 = 400
(-1)*(-400) = 400
(-2)*(-200) = 400
(-4)*(-100) = 400
(-5)*(-80) = 400
(-8)*(-50) = 400
(-10)*(-40) = 400
(-16)*(-25) = 400
(-20)*(-20) = 400


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 40:



First NumberSecond NumberSum
14001+400=401
22002+200=202
41004+100=104
5805+80=85
8508+50=58
104010+40=50
162516+25=41
202020+20=40
-1-400-1+(-400)=-401
-2-200-2+(-200)=-202
-4-100-4+(-100)=-104
-5-80-5+(-80)=-85
-8-50-8+(-50)=-58
-10-40-10+(-40)=-50
-16-25-16+(-25)=-41
-20-20-20+(-20)=-40




From the table, we can see that the two numbers 20 and 20 add to 40 (the middle coefficient).



So the two numbers 20 and 20 both multiply to 400 and add to 40



Now replace the middle term 40x with 20x%2B20x. Remember, 20 and 20 add to 40. So this shows us that 20x%2B20x=40x.



x%5E2%2Bhighlight%2820x%2B20x%29%2B400 Replace the second term 40x with 20x%2B20x.



%28x%5E2%2B20x%29%2B%2820x%2B400%29 Group the terms into two pairs.



x%28x%2B20%29%2B%2820x%2B400%29 Factor out the GCF x from the first group.



x%28x%2B20%29%2B20%28x%2B20%29 Factor out 20 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%2B20%29%28x%2B20%29 Combine like terms. Or factor out the common term x%2B20



%28x%2B20%29%5E2 Condense the terms.



===============================================================



Answer:



So x%5E2%2B40%2Ax%2B400 factors to %28x%2B20%29%5E2.



In other words, x%5E2%2B40%2Ax%2B400=%28x%2B20%29%5E2.



Note: you can check the answer by expanding %28x%2B20%29%5E2 to get x%5E2%2B40%2Ax%2B400 or by graphing the original expression and the answer (the two graphs should be identical).