SOLUTION: Factor the following polynomial completely. 25a4 + 40a2 + 16 =

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Question 202196: Factor the following polynomial completely.
25a4 + 40a2 + 16 =
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

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Looking at 25a%5E4%2B40a%5E2%2B16 we can see that the first term is 25a%5E4 and the last term is 16 where the coefficients are 25 and 16 respectively.

Now multiply the first coefficient 25 and the last coefficient 16 to get 400. Now what two numbers multiply to 400 and add to the middle coefficient 40? Let's list all of the factors of 400:



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

-1,-2,-4,-5,-8,-10,-16,-20,-25,-40,-50,-80,-100,-200 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 400
1*400
2*200
4*100
5*80
8*50
10*40
16*25
20*20
(-1)*(-400)
(-2)*(-200)
(-4)*(-100)
(-5)*(-80)
(-8)*(-50)
(-10)*(-40)
(-16)*(-25)
(-20)*(-20)

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


Now which of these pairs add to 40? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 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 this list we can see that 20 and 20 add up to 40 and multiply to 400


Now looking at the expression 25a%5E4%2B40a%5E2%2B16, replace 40a%5E2 with 20a%5E2%2B20a%5E2 (notice 20a%5E2%2B20a%5E2 adds up to 40a%5E2. So it is equivalent to 40a%5E2)

25a%5E4%2Bhighlight%2820a%5E2%2B20a%5E2%29%2B16


Now let's factor 25a%5E4%2B20a%5E2%2B20a%5E2%2B16 by grouping:


%2825a%5E4%2B20a%5E2%29%2B%2820a%5E2%2B16%29 Group like terms


5a%5E2%285a%5E2%2B4%29%2B4%285a%5E2%2B4%29 Factor out the GCF of 5a%5E2 out of the first group. Factor out the GCF of 4 out of the second group


%285a%5E2%2B4%29%285a%5E2%2B4%29 Since we have a common term of 5a%5E2%2B4, we can combine like terms


So 25a%5E4%2B20a%5E2%2B20a%5E2%2B16 factors to %285a%5E2%2B4%29%285a%5E2%2B4%29


So this also means that 25a%5E4%2B40a%5E2%2B16 factors to %285a%5E2%2B4%29%285a%5E2%2B4%29 (since 25a%5E4%2B40a%5E2%2B16 is equivalent to 25a%5E4%2B20a%5E2%2B20a%5E2%2B16)


note: %285a%5E2%2B4%29%285a%5E2%2B4%29 is equivalent to %285a%5E2%2B4%29%5E2 since the term 5a%5E2%2B4 occurs twice. So 25a%5E4%2B40a%5E2%2B16 also factors to %285a%5E2%2B4%29%5E2



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Answer:
So 25a%5E4%2B40a%5E2%2B16 factors to %285a%5E2%2B4%29%5E2


In other words, 25a%5E4%2B40a%5E2%2B16=%285a%5E2%2B4%29%5E2

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