SOLUTION: a^4-15a^2+56

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Question 111328: a^4-15a^2+56
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Looking at a%5E4-15a%5E2%2B56 we can see that the first term is a%5E4 and the last term is 56 where the coefficients are 1 and 56 respectively.

Now multiply the first coefficient 1 and the last coefficient 56 to get 56. Now what two numbers multiply to 56 and add to -15? Let's list all of the factors of 56:



Factors of 56:
1,2,4,7,8,14,28,56

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

These factors pair up and multiply to 56
1*56
2*28
4*14
7*8
(-1)*(-56)
(-2)*(-28)
(-4)*(-14)
(-7)*(-8)

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


Now which of these pairs add to -15? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -15

First NumberSecond NumberSum
1561+56=57
2282+28=30
4144+14=18
787+8=15
-1-56-1+(-56)=-57
-2-28-2+(-28)=-30
-4-14-4+(-14)=-18
-7-8-7+(-8)=-15



From this list we can see that -7 and -8 add up to -15 and multiply to 56


Now looking at the expression a%5E4-15a%5E2%2B56, replace -15a%5E2 with -7a%5E2%2B-8a%5E2 (notice -7a%5E2%2B-8a%5E2 adds up to -15a%5E2. So it is equivalent to -15a%5E2)

a%5E4%2Bhighlight%28-7a%5E2%2B-8a%5E2%29%2B56


Now let's factor a%5E4-7a%5E2-8a%5E2%2B56 by grouping:


%28a%5E4-7a%5E2%29%2B%28-8a%5E2%2B56%29 Group like terms


a%5E2%28a%5E2-7%29-8%28a%5E2-7%29 Factor out the GCF of a%5E2 out of the first group. Factor out the GCF of 8 out of the second group


So this also means that a%5E4-15a%5E2%2B56 factors to %28a%5E2-8%29%28a%5E2-7%29 (since a%5E4-15a%5E2%2B56 is equivalent to a%5E4-7a%5E2-8a%5E2%2B56)