SOLUTION: 64a^4-224a^2+196=

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Question 251559: 64a^4-224a^2+196=
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

64a%5E4-224a%5E2%2B196 Start with the given expression


4%2816a%5E4-56a%5E2%2B49%29 Factor out the GCF 4


Now let's focus on the inner expression 16a%5E4-56a%5E2%2B49




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Looking at 16a%5E4-56a%5E2%2B49 we can see that the first term is 16a%5E4 and the last term is 49 where the coefficients are 16 and 49 respectively.

Now multiply the first coefficient 16 and the last coefficient 49 to get 784. Now what two numbers multiply to 784 and add to the middle coefficient -56? Let's list all of the factors of 784:



Factors of 784:
1,2,4,7,8,14,16,28,49,56,98,112,196,392

-1,-2,-4,-7,-8,-14,-16,-28,-49,-56,-98,-112,-196,-392 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 784
1*784
2*392
4*196
7*112
8*98
14*56
16*49
28*28
(-1)*(-784)
(-2)*(-392)
(-4)*(-196)
(-7)*(-112)
(-8)*(-98)
(-14)*(-56)
(-16)*(-49)
(-28)*(-28)

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


Now which of these pairs add to -56? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -56

First NumberSecond NumberSum
17841+784=785
23922+392=394
41964+196=200
71127+112=119
8988+98=106
145614+56=70
164916+49=65
282828+28=56
-1-784-1+(-784)=-785
-2-392-2+(-392)=-394
-4-196-4+(-196)=-200
-7-112-7+(-112)=-119
-8-98-8+(-98)=-106
-14-56-14+(-56)=-70
-16-49-16+(-49)=-65
-28-28-28+(-28)=-56



From this list we can see that -28 and -28 add up to -56 and multiply to 784


Now looking at the expression 16a%5E4-56a%5E2%2B49, replace -56a%5E2 with -28a%5E2-28a%5E2 (notice -28a%5E2-28a%5E2 combines back to -56a%5E2. So it is equivalent to -56a%5E2)

16a%5E4%2Bhighlight%28-28a%5E2-28a%5E2%29%2B49


Now let's factor 16a%5E4-28a%5E2-28a%5E2%2B49 by grouping:


%2816a%5E4-28a%5E2%29%2B%28-28a%5E2%2B49%29 Group like terms


4a%5E2%284a%5E2-7%29-7%284a%5E2-7%29 Factor out the GCF of 4a%5E2 out of the first group. Factor out the GCF of -7 out of the second group


%284a%5E2-7%29%284a%5E2-7%29 Since we have a common term of 4a%5E2-7, we can combine like terms

So 16a%5E4-28a%5E2-28a%5E2%2B49 factors to %284a%5E2-7%29%284a%5E2-7%29


So this also means that 16a%5E4-56a%5E2%2B49 factors to %284a%5E2-7%29%284a%5E2-7%29 (since 16a%5E4-56a%5E2%2B49 is equivalent to 16a%5E4-28a%5E2-28a%5E2%2B49)


note: %284a%5E2-7%29%284a%5E2-7%29 is equivalent to %284a%5E2-7%29%5E2 since the term 4a%5E2-7 occurs twice. So 16a%5E4-56a%5E2%2B49 also factors to %284a%5E2-7%29%5E2



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So our expression goes from 4%2816a%5E4-56a%5E2%2B49%29 and factors further to 4%284a%5E2-7%29%5E2


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Answer:

So 64a%5E4-224a%5E2%2B196 factors to 4%284a%5E2-7%29%5E2