SOLUTION: Factor 64a^2-16ab+b^2

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Question 610313: Factor 64a^2-16ab+b^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Looking at the expression 64a%5E2-16ab%2Bb%5E2, we can see that the first coefficient is 64, the second coefficient is -16, and the last coefficient is 1.


Now multiply the first coefficient 64 by the last coefficient 1 to get %2864%29%281%29=64.


Now the question is: what two whole numbers multiply to 64 (the previous product) and add to the second coefficient -16?


To find these two numbers, we need to list all of the factors of 64 (the previous product).


Factors of 64:
1,2,4,8,16,32,64
-1,-2,-4,-8,-16,-32,-64


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


These factors pair up and multiply to 64.
1*64 = 64
2*32 = 64
4*16 = 64
8*8 = 64
(-1)*(-64) = 64
(-2)*(-32) = 64
(-4)*(-16) = 64
(-8)*(-8) = 64

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -16:


First NumberSecond NumberSum
1641+64=65
2322+32=34
4164+16=20
888+8=16
-1-64-1+(-64)=-65
-2-32-2+(-32)=-34
-4-16-4+(-16)=-20
-8-8-8+(-8)=-16



From the table, we can see that the two numbers -8 and -8 add to -16 (the middle coefficient).


So the two numbers -8 and -8 both multiply to 64 and add to -16


Now replace the middle term -16ab with -8ab-8ab. Remember, -8 and -8 add to -16. So this shows us that -8ab-8ab=-16ab.


64a%5E2%2Bhighlight%28-8ab-8ab%29%2Bb%5E2 Replace the second term -16ab with -8ab-8ab.


%2864a%5E2-8ab%29%2B%28-8ab%2Bb%5E2%29 Group the terms into two pairs.


8a%288a-b%29%2B%28-8ab%2Bb%5E2%29 Factor out the GCF 8a from the first group.


8a%288a-b%29-b%288a-b%29 Factor out the -b from the second group.


%288a-b%29%288a-b%29 Factor out 8a-b from the entire expression.

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Answer:


So 64a%5E2-16ab%2Bb%5E2 factors to %288a-b%29%288a-b%29.


In other words, 64a%5E2-16ab%2Bb%5E2=%288a-b%29%288a-b%29 for all values of 'a' and 'b'.


Note: you can check the answer by expanding %288a-b%29%288a-b%29 to get 64a%5E2-16ab%2Bb%5E2 back again or by graphing the original expression and the answer (the two graphs should be identical).